On growth and form

GROWTH AND FORM

Ah instantdnedus photograph of a 'splash' of milk. From Harold E.
' '"*' KdgertbTiTirlassachusetts Institute of Technology

See p. 390

ON 94

GROWTH AND FORM ^^

BY

D'ARCY WENTWORTH THOMPSON

A new edition

MARINE

BIOLOGICAL

LABORATORY

LIBRARY

WOODS HOLE, MASS.
W. H. 0. 1.

CAMBRIDGE: AT THE UNIVERSIT
NEW YORK: THE MACMILLAN CC

Y PRESS
)MPANY

1945

"The reasonings about the wonderful and intricate operations
of Nature are so full of uncertainty, that, as the Wise-man truly
observes, hardly do we guess aright at the things jhat are upon
earth, and with labour do we find the things that are before us.''
Stephen Hales, Vegetable Staticks (1727), p. 318, 1738.

"Ever since I have been enquiring into the works of Nature
I have always loved and admired the Simplicity of her Ways."
Dr George Martine (a pupil of Boorhaave's), in Medical Essays and
Observations, Edinburgh, 1747.

PREFATORY NOTE

THIS book of mine has little need of preface, for indeed it is
"all preface" from beginning to end. I have written it as
an easy introduction to the study of organic Form, by methods
which are the common-places of physical science, which are by
no means novel in their application to natural history, but which
nevertheless naturalists are little accustomed to employ.

It is not the biologist with an inkling of mathematics, but
the skilled and learned mathematician who must ultimately deal
with such problems as are sketched and adumbrated here. I pretend
to no mathematical skill, but I have made what use I could of
what tools I had; I have dealt with simple cases, and the mathe-
matical methods which I have introduced are of the easiest and
simplest kind. Elementary as they are, my book has not been
written without the help — the indispensable help^ — of many friends.
Like Mr Pope translating Homer, when I felt myself deficient I
sought assistance! And the experience which Johnson attributed
to Pope has been mine also, that men of learning did not refuse
to help me.

I wrote this book in wartime, and its revision has employed
me during another war. It gave me solace and occupation, when
service was debarred me by my years.

Few are left of the friends who helped me write it, but I do not
forget the debt I owe them all. Let me add another to these
kindly names, that of Dr G. T. Bennett, of Emmanuel College,
Cambridge; he has never wearied of collaboration with me, and
his criticisms have been an education to receive.

D. W. T.
1916-1941.

American edition published August, 1942

Reprinted January, 1943

Reprinted May, 1944

Reprinted May, 1945

PRINTED IN THE UNITED STATES OF AMERICA

CONTENTS

CHAP. PAGE

I. Introductory 1

II. On Magnitude 22

III. The Rate of Growth 78

IV. On the Internal Form and Structure of the Cell . . 286

V. The Forms of Cells 346

VI. A Note on Adsorption 444

VII. The Forms of Tissues, or Cell-aggregates . . . 465

VIII. The same (continued) . . 566

IX. On Concretions, Spicules, and Spicular Skeletons . . 645

X. A Parenthetic Note on Geodetics 741

XI. The Equiangular Spiral 748

XII. The Spiral Shells of the Foraminifera .... 850

XIII. The Shapes of Horns, and of Teeth or Tusks: with

a Note on Torsion 874

XIV. On Leaf-arrangement, or Phyllotaxis .... 912

XV. On the Shapes of Eggs, and of certain other Hollow

Structures 934,

XVI. On Form and Mechanical Efficiency .... 958

XVII. On the Theory of Transformations, or the Comparison

OF Related Forms 1026

Epilogue 1093

Index 1095

Plates

A Splash of Milk Frontispiece

The Latter Phase of a Splash . . " . . . . facing page 390

"The mathematicians are well acquainted with the difference
between pure science, which has to do only with ideas, and the
application of its laws to the use of life, in which they are con-
strained to submit to the imperfections of matter and the influence
of accident." Dr Johnson, in the fourteenth Rambler, May 5, 1750.

"Natural History. . .is either the beginning or the end of physical
science." Sir John Herschel, in The Study of Natvml Philosophy,
p. 221, 1831.

"I believe the day must come when the biologist will — without
being a mathematician — not hesitate to use mathematical analysis
when he requires it." Karl Pearsonf in Nature, January 17, 1901.

CHAPTER I

INTRODUCTORY

Of the chemistry of his day and generation, Kant declared that it
was a science, but not Science — eine Wissenschaft, aber nicht Wissen-
schaft — for that the criterion of true science lay in its relation to
mathematics*. This was an old story : for Roger Bacon had called
mathematics porta et clavis scientiarum, and Leonardo da Vinci had
said much the samef. Once again, a hundred years after Kant,
Du Bois Reymond, profound student of the many sciences on which
physiology is based, recalled the old saying, and declared that
chemistry would only reach the rank of science, in the high and
strict sense, when it should be found possible to explain che^dcal
reactions in the light of their causal relations to the velocities,
tensions and conditions of equihbrium of the constituent molecules ;
that, in short, the chemistry of the future must deal with molecular
mechanics by the methods and in the strict language of mathematics,
as the astronomy of Newtoii and Laplace dealt with the stars in
their courses. We know how great a step was made towards this
distant goal as Kant defined it, when van't Hoif laid the firm
foundations of a mathematical chemistry, and earned his proud
epitaph — Physicam chemiae adiunxitX-

We need not wait for the full reahsation of Kant's desire, to apply
to the natural sciences the principle which he laid down. Though
chemistry fall short of its ultimate goal in mathematical mechanics §,
nevertheless physiology is vastly strengthened and enlarged by

* " Ich behaupte nur dass in jeder besonderen Naturlehre nur so viel eigentliche
Wissenschaft angetroffen konne als darin Mathematik anzutreffen ist" : Gesammelte
Schriften, iv, p. 470.

t "Nessuna humana investigazione si puo dimandare vera scienzia s'essa non
passa per le matematiche dimostrazione."

X Cf. also Crum Brown, On an application of Mathematics to Chemistry, Trans.
R.S.E. XXIV, pp. 691-700, 1867.

§ Ultimate, for, as Francis Bacon tells us: Mathesis philosophiam naturalem
terminare debet, non generare aut procreare.

2 INTRODUCTORY <^i,J^^H

making use of the chemistry, and of the physics, of the age. Littl
by httle it draws nearer to our conception of a true science with
each branch of physical science which it brings into relation with
itself: with every physical law and mathematical theorem which it
learns to take into its employ*. Between the physiology of Haller,
fine as it was, and that of Liebig, Helmholtz, Ludwig, Claude
Bernard, there was all the difference in the worldf.

As soon as we adventure on the paths of the physicist, we learn
to weigh and to measure, to deal with time and space and mass and
their related concepts, and to find more and more our knowledge
expressed and our needs satisfied through the concept of number,
as in the dreams and visions of Plato and Pythagoras ; for modern
chemistry would have gladdened the hearts of those great philo-
sophic dreamers. Dreams apart, numerical precision is the very
soul of science, and its attainment affords the best, perhaps, the
only criterion of the truth of theories and the correctness of experi-
mentsj:. So said Sir John Herschel, a hundred years ago; and
Kant had said that it was Nature herself, and not the mathematician,
who brings mathematics into natural philosophy.

But the zoologist or morphologist has been slow, where the
physiologist has long been eager, to invoke the aid of the physical
or mathematical sciences; and the reasons for this difference lie
deep, and are partly rooted in old tradition and partly in the
diverse minds and temperaments of men. To treat the living body
as a mechanism was repugnant, and seemed even ludicrous, to
Pascal §; and Goethe, lover of nature as he was, ruled mathematics
out of place in natural history. Even now the zoologist has scarce
begun to dream of defining in mathematical language even the
simplest organic forms. When he meets with a simple geometrical

* "Sine profunda Mechanices Scientia nil veri vos intellecturos, nil boni pro-
laturos aliis": Boerhaave, De usu ratiocinii Mechanici in Medicina, 1713.

t It is well within my own memor how Thomson and Tait, and Klein and
Sylvester had to lay stress on the mathematical aspect, and urge the mathematical
study, of physical science itself!

X Dr Johnson says that "to count is a modern practice, the ancient method was
to guess"; but Seneca was alive to the difference — "magnum esse solem philosophus
probabit, quantus sit mathematicus."

§ Cf. Pensees, xxix, "II faut dire, en gros, cela se fait par figure et mouvement,
car cela est yrai. Mais de dire quels, et composer la machine, cela est ridicule,
car cela est inutile, et incertain, et penible."

I] OF ADAPTATION AND FITNESS 3

construction, for instance in the honeycomb, he would fain refer it
to psychical instinct, or to skill and ingenuity, rather than to the
operation of physical forces or mathematical laws; when he sees in
snail, or nautilus, or tiny foraminiferal or radiolarian shell a close
approach to sphere or spiral, he is prone of old habit to believe that
after all it is something more than a spiral or a sphere, and that in
this "something more" there lies what neither mathematics nor
physics can explain. In short, he is deeply reluctant to compare
the living with the dead, or to explain by geometry or by mechanics
thp things which have their part in the mystery of life. Moreover
he IS httle inclined to feel the need of such explanations, or of such
extension of his field of thought. He is not without some justifi-
cation if he feels that in admiration of nature's handiwork he has
an horizon open before his eyes as wide as any man requires. He
has the help of many fascinating theories within the bounds of his
own science, which, though a little lacking in precision, serve the
purpose of ordering his thoughts and of suggesting new objects of
enquiry. His art of classification becomes an endless search after
the blood-relationships of things living and the pedigrees of things
dead and gone. The facts of embryology record for him (as Wolff,
von Baer and Fritz Miiller proclaimed) not only the life-history of
the individual but the ancient annals of its race. The facts of
geographical distribution or even of the migration of birds lead on
and on to speculations regarding lost continents, sunken islands, or
bridges across ancient seas. Every nesting bird, every ant-hill or
spider's web, displays its psychological problenis of instinct or intel-
ligence. Above all, in things both great and small, the naturajist
is rightfully impressed and finally engrossed by the peculiar beauty
which is manifested in apparent fitness or "adaptation" — the flower
for the bee, the berry for the bird.

Some lofty concepts, like space and number, involve truths remote
from the category of causation; and here we must be content, as
Aristotle says, if the mere facts be known*. But natural history
deals with ephemeral and accidental, not eternal nor universal

* ovK diraiTrjTeop 5 ov8i ttju airiav ofxoiojs, dW LKavou ^v tlctl rb on deLxdrjvai /caXcDj
Eth. Nic. 1098a, 33. Teleologist as he was at heart, Aristotle realised that mathematics
was on another plane to teleology: rds 5^ fxad-qfiariKas oOdeva iroieladai \6you wepi
dyaduv xat KaKQv. Met. 996a, 35.

4 INTRODUCTORY [ch.

things ; their causes and effects thru^ themselves on our curiosity, and
become the ultimate relations to which our contemplation extends*.

Time out of mind it has been by way of the "final cause," by the
teleological concept of end, of purpose or of "design," in one of its
many forms (for its moods are many), that men have been chiefly
wont to explain the phenomena of th« Hving world ; and it will be
so while men have eyes to see and ears to hear withal. With Galen,
as with Aristotlef, it was the physician's way; with John Ray J, as
with Aristotle, it was the naturahst's way; with Kant, as with
Aristotle, it was the philosopher's way. It was the old Hebrew
way, and has its splendid setting in the story that God made "every
plant of the field before it was in the earth, and every herb of the
field before it grew." It is a common way, and a great way; for it
brings with it a glimpse of a great vision, and it hes deep as the
love of nature in the hearts of men.

The argument of the final cause is conspicuous in eighteenth-
century physics, half overshadowing the "efficient" or physical
cause in the hands of such men as Euler§, or Fermat or Maupertuis,
to whom Leibniz 1 1 had passed it on. Half overshadowed by the
mechanical concept, it runs through Claude Bernard's Legons sur les
phenomenes de la Vie^, and abides in much of modern physiology**.

* "All reasonings concerning matters of fact seem to be founded on the relation
of Cause and Effect. By means of that relation alone we go beyond the evidence
of our memory and senses": David Hume, On the Operations of the Understanding.

t E.g. "In the works of Nature purpose, not accident, is the main thing": to yap
fir) TvxofTWs, d\\' eveKOL rtros, ev tols tt]S (pixreajs ^pyoLs ecrl ko-I jxaXicra. PA, 645a, 24.

X E.g. "Quaeri fortasse a nonnullis potest, Quis Papilionum usus? Respondeo,
ad ornatum Universi, et ut hominibus spectaculo sint." Joh. Rail, Hist. Insedorum,
p. 109.

§ "Quum enim Mundi universi fabrica sit perfectissima, atque a Creators
sapientissimo absoluta, nihil omnino in Mundo contingit in quo non maximi
minimive ratio quaepiam eluceat; quamobrem dubium prorsus est nullum quin
omnes Mundi effectus ex causis finalibus, ope Methodi maximorum et minimorum,
aeque feliciter determinari queant atque ex ipsis causis efficientibus." Methodus
inveniendi, etc., 1744, p. 24o {cit. Mach, Science of Mechanics, 1902, p. 455).

!| Cf. Opera (ed. Erdmann), p. 106, "Bien loin d'exclure les causes finales...
c'est de la qu'il faut tout deduire en Physique": in sharp contrast to Descartes's
teaching, " NuUas unquam res naturales a fine, quem Deus aut Natura in iis faciendis
sib jproposuit, desumemus, etc." Princip. i, 28.

Tj Cf. p. 162. "La force vitale dirige des phenomenes qu'elle ne produit pas:
les agents physiques produisent des phenomenes qu'ils ne dirigent pas."

** It is now and then conceded with reluctance. Thus Paolo Enriques, a learned
and philosophic naturalist, writing '"dell' economia di sostanza nelle osse cave"

I] OF THE FINAL CAUSE 5

Inherited from Hegel, it dominated Oken's Naturphilosophie and

lingered among his later disciples, who were wont to liken the course

of organic evolution not to the straggling branches of a tree, but to

the building of a temple, divinely planned, and the crowning of it

with its pohshed minarets*.

It is retained, somewhat crudely, in modern embryology, by those

who see in th^ early processes of growth a significance ''rather

prospective than retrospective," such that the enlbryonic phenomena

must "be referred directly to their usefulness in building up the

body of the future animalf": — which is no more, and no less, than

to say, with Aristotle, that the organism is the reAo?, or final cause,

of its own processes of generation and development. It is writ

large in that Entelechyij: which Driesch rediscovered, and which he

made known to many who had neither learned of it from Aristotle,

nor studied it with Leibniz, nor laughed at it with Rabelais and

Voltaire. And, though it is in a very curious way, we are told

that teleology was "refounded, reformed and rehabihtated " by

Darwin's concept of the origin of species §; for, just as the older

naturalists held (as Addison || puts it) that "the make of every kind

of animal is different from that of every other kind; and ye. ti^ere

is not the least turn in the muscles, or twist in the fibres of any one,

which does not render them more proper for that particular animal's

way of life than any other cut or texture of them would have been " :

so, by the theory of natural selection, "every variety of form and

colour was urgently and absolutely called upon to produce its title

{Arch. f. Entw. Mech. xx, 1906), says "una certa impronta di teleologismo qua
e la e rimasta, mio malgrado, in questo scritto."

* Cf. John Cleland, On terminal forms of life, Journ. Anat. and Physiol.
XVIII, 1884.

t Conklin, Embryology of Crepidula, Journ. of Morphol. xin, p. 203, 1897;
cf. F. R. Lillie, Adaptation in cleavage. Wood's Hole Biol. Lectures, 1899, pp. 43-67.

X I am inclined to trace back Driesch's teaching of Entelechy to no less a person
than Melanchthon. When Bacon {de Augm. iv, 3) states with disapproval that
the soul "has been regarded rather as a function than as a substance," Leslie
Ellis points out that he is referring to Melanchthon's exposition of the Aristotelian
doctrine. For Melanchthon, whose view of the peripatetic philosophy had great
and lasting influence in the Protestant Universities, affirmed that, according to
the true view of Aristotle's opinion, the soul is not a substance but an ivreX^xeia, or
function. He defined it as Sufa/xis quaedam ciens actiones — a description all but
identical with that of Claude Bernard's '''force vitale.''

§ Ray Lankester, art. Zoology, Encycl. Brit. (9th edit.), 1888, p. 806.

!l Spectator, No. 120.

6 INTRODUCTORY [ch.

to existence either as an active useful agent, or as a survival" of
such active usefulness in the past. But in this last, and very
important case, we have reached a teleology without a rdXos, as
men like Butler and Janet have been prompt to shew, an "adapta-
tion" without "design," a teleology in which the final cause becomes
little more, if anything, than the mere expression or resultant of a
sifting out of the good from the bad, or of the better from the worse,
in short of a process of mechanism. The apparent manifestations
of purpose or adaptation become part of a mechanical philosophy,
"une forme methodologique de connaissance*," according to which
"la Nature agit tou jours par les moyens les plus simplest," and
"chaque chose finit toujours par s'accommoder a son miUeu," as in
the Epicurean creed or aphorism that ^atnTe finds a use for every-
thing J. In short, by a road which resembles but is not the same as
Maupertuis's road, we find our way to the very world in which we
are living, and find that, if it be not, it is ever tending to become,
"the best of all possible worlds §."

But the use of the teleological principle is but one way, not the
whole or the only way, by which we may seek to learn how things
came to be, and to take their places in the harmonious complexity
of the world. To seek not for end^ but for antecedents is the way
of the physicist, who finds "causes" in what he has learned to
recognise as fundamental properties, or inseparable concomitants,
or unchanging laws, of matter and of energy. In Aristotle's parable,
the house is there that men may live in it ; but it is also there because
the builders have laid one stone upon another. It is as a mechanism,
or a mechanical construction, that the physicist looks upon the
world; and Democritus, first of physicists and one of the greatest
of the Greeks, chose to refer all natural phenomena to mechanism
and set the final cause aside.

* So Newton, in the Preface to the Principia: "Natura enim simplex est, et
rerura causis superfluis non -luxuriat"; "Nature is pleased with simplicity, and
affects not the pomp of superfluous causes." Modern physics finds the perfection
of mathematical beauty in what Newton called the perfection of simplicity.

t Janet, Les Causes Finales, 1876, p. 350.

X "Nil ideo quoniam natumst in corpore ut uti Possemus sed quod natumst id
procreat usum." Lucret. iv, 834.

§ The phrase is Leibniz's, in his Theodicee: and harks back to Aristotle — If one
way be better than another, that you may be sure is Nature's way; Nic. Eth.
10996, 23 et al.

I] OF EFFICIENT AND FINAL CAUSES 7

Still, all the whiLe, like warp and woof, mechanism and teleology
are interwoven together, and we must not cleave to the one nor
despise the other; for their union is rooted in the very nature of
totality. We may grow shy or weary of looking to a final cause
for an explanation of our phenomena ; but after we have accounted
for these on the plainest principles of mechanical causation it may
be useful and appropriate to see how the final cause would tally
wuth the other, and lead towards the same conclusion*. Maupertuis
had Uttle hking for the final cause, and shewed some sympathy with
Descartes in his repugnance to its appHcation to physical science.
But he found at last, taking the final and the efficient causes one with
another, that "I'harmonie de ces deux attributs est si parfaite que
sans doute tous les effets de la Nature se pourroient deduire de
chacun pris separement. Une Mecanique aveugle et necessaire suit
les dessins de I'lntelHgence la plus eclairee et la plus hbref." Boyle
also, the Father of Chemistry, wrote, in his latter years, a Disquisition
about the Final Causes of Natural Things: Wherein it is Inquired
Whether, And {if at all) With what Cautions, a Naturalist should admit
Themi He found "that all consideration of final cause is not to be
banished from- Natural Philosophy..."; but on the other hand
''that the naturahst who would deserve that name must not let
the search and knowledge of final causes make him neglect the in-
dustrious indagation of efficients J." In our own day the philosopher
neither minimises nor unduly magnifies the mechanical aspect of
the Cosmos; nor need the naturahst either exaggerate • or be-
little the mechanical phenomena which are profoundly associated
with Life, and inseparable from our understanding of Growth and
Form. ^

* "S'il est dangereux de se servir des causes finales a priori pour trouver les lois
des phenomenes, 11 est peut-etre utile et il est au moins curieux de faire voir com-
ment le principe des causes finales s'accorde avec les lois des phenomenes, pourvu
qu'on commence par determiner ces lois d'apres les principes de mecanique clairs
et incontestables." (D'Aiembert, Art. Causes finales, Encyclopedie, ii, p. 789, 1751.)
I SeeJiis essay on the '''Accord des differentes lois de la Nature."
X Cf. also Leibniz {Discours de la Metaphysique: Lettres inedites, ed. de Careil,
1857, p. 354), "L'un et I'autre est bon, I'un et I'autre peut etre utile... et les
auteurs qui suivent ces deux routes differentes ne devraient pas se maltraiter."
Or again in the Monadologie, "Les ames agissent selon les causes finales. ... Les
corps agissent selon les lois des causes efficientes ou des mouveraents. Et les
deux regnes, celui des causes efficientes et des causes finales sont harmonieux
entre eux."

8 INTRODUCTORY [ch.

Nevertheless, when philosophy bids us hearken and obey the
lessons both of mechanical and of teleological interpretation, the
precept is hard to follow: so that oftentimes it has come to pass,
just as in Bacon's day, that a leaning to the side of the final cause
"hath intercepted the severe and diligent enquiry of all real and
physical causes," and has brought it about that "the search of the
physical cause hath been neglected and passed in silence." So long
and so far as "fortuitous variation*" and the "survival of the
fittest" remain engrained as fundamental and satisfactory hypo-
theses in the philosophy of biology, so long will these "satisfactory
and specious causes" tend to stay "severe and diligent enquiry. . .
to the great arrest and prejudice of future discovery." Long
before the great Lord Keeper wrote these words, Roger Bacon had
shewn how easy it is, and how vain, to survey the operations of
Nature and idly refer her wondrous works to chance or accident,
or to the immediate interposition of Godf.

The difiiculties which surround the concept of ultimate or "real"
caugation, in Bacon's or Newton's sense of the word, the in-
superable difficulty of giving any just and tenable account of the
relation of cause and effect from the empirical point of view, need
scarcely hinder us in our physical enquiry. As students of mathe-
matical and experimental physics we are content to deal with those
antecedents, or concomitants, of our phenomena without which the
phenomenon does not occur — with causes, in short, which, aliae ex
aliis aptaetet necessitate' nexae, are no more, and no less, than con-
ditions sine qua non. Our purpose is still adequately . fulfilled :
inasmuch as we are still enabled to correlate, and to equate, our
particular phenomena with more and more of the physical phenomena
around, and so to weave a web of connection and interdependence
which shall serve our turn, though the metaphysician withhold from
that interdependence the title of causality J. We come in touch

* The reader will understand that I speak, not of the "severe and diligent
enquiry" of variation or of fortuity, but merely of the easy assumption that these
phenomena are a sufficient basis on which to rest, with the all-powerful help of
natural selection, a theory of definite and progressive evolution.

f Op. tert. (ed. Brewer, p. 99). "Ideo mirabiles actiones naturae, quae tota
die fiunt in nobis et in rebus coram oculis nostris, non percipimus; sed aestimamus
eas fieri vel per specialem operationem divinam. . .vel a casu et fortuna,"

t Cf. Fourier's phrase, in his Theorie de la Chaleur, with which Thomson and
Tait prefaced their Treatise an Natural Philosophy: "Les causes primordiales ne

I] OF ULTIMATE CAUSATION 9

with what the schoohnen called a ratio cognoscendi, though the true
ratio efflciendi is still enwrapped in many mysteries. And so handled,
the quest of physical causes merges with another great Aristotelian
theme — the search for relations between things apparently dis-
connected, and for "similitude in things to common view unlike*."
Newton did not shew the cause of the apple falling, but he shewed
a simihtude ("the more to increase our wonder, with an apple")
between the apple and the starsf. By doing so he turned old facts
into new knowledge ; and was well content if he could bring diverse
phenomena under "two or three Principles of Motion" even "though
the Causes of these Principles were not yet discovered".

Moreover, the naturalist and the physicist will continue to speak
of "causes", just as of old, though it may be with some mental
reservations : for, as a French philosopher said in a kindred difficulty :
"ce sont la des manieres de s'exprimer, et si elles sont interdites
il faut renoncer a parler de ces choses."

The search for differences or fundamental contrasts between the
phenomena of organic and inorganic, of animate and inanimate,
things,, has occupied many men's minds, while the search for com-
munity of principles or essential simihtudes has been pursued by
few; and the contrasts are apt to loom too large, great though they
may be. M. Dunan, discussing the Probleme de la Viel, in an essay
which M. Bergson greatly commends, declares that "les lois physico-
chimiques sont aveugles et brutales ; la ou elles regnent seules, au
lieu d'un ordre et d'un concert, il ne pent y avoir qu'incoherence et
chaos." But the physicist proclaims aloud that the physical
phenomena which meet us by the way have their forms not less
beautiful and scarce less varied than those which move us to admira-

nous sont point connues; mais elles sont assujetties a des lois simples et eonstantes,
que Ton peut decouvrir par I'observation, et dont I'etude est I'objet de la philosophie
naturelle."

* "Plurimum amo analogias, fidelissimos meos magistros, omnium Naturae
arcanorum conscios," said Kepler; and Perrin speaks with admiration, in Les
Atonies, of men like Galileo and Carnot, who "possessed the power of perceiving
analogies to an extraordinary degree." Hure^e declared, and Mill said much the
same thing, that all reasoning whatsoever depends on resemblance or analogy,
and the power to recognise it. Comparative anatomy (as Vicq d'Azyr first called
it), or comparative physics (to use a phrase of Mach's), are particular instances of
a sustained search for analogy or similitude.

t As for Newton's apple, see De Morgan, in Notes and Queries (2), vi, p. 169, 1858.

I Revue Philosophique, xxxiii, 1892.

10 INTRODUCTORY [ch.

tion among living things. The waves of the sea, the little ripples
on the shore, the sweeping curve of the sandy bay between the
headlands, the outline of the hills, the shape of the clouds, all these
are so many riddles of form, so many problems of morphology, and
all of them the physicist can more or less easily read and adequately
solve: solving them by reference to their antecedent phenomena,
in the material system of mechanical forces to which they belong,
and to which we interpret them as being due. They have also,
doubtless, their immanent teleological significance; but it is on
another plane of thought from the physicist's that we contemplate
their intrinsic harmony* and perfection, and "see that they are
good."

Nor is it otherwise with the material forms of living things. Cell
and tissue, shell and bone, leaf and flower, are so many portions of
matter, and it is in obedience to the laws of physics that their
particles have been moved, moulded and conformedf. They are no
exception to the rule that Qeos aet yeajjjLerpeL. Their problems of
form are in the first instance mathematical problems, their problems
of growth are essentially physical problems, and the morphologist is,
ipso facto, a student of physical science. He may learn from that
comprehensive science, as the physiologists have not failed to do,
the point of view from which her problems are approached, the
quantitative methods by which they are attacked, and the whole-
some restraints under which all her work is done. He may come
to realise that there is no branch of mathematics, however abstract,
which may not some day be applied to phenomena of the real

* What I understand by "holism" is what the Greeks called apfiovia. This is
something exhibited not only by a lyre in tune, but by all the handiwork of
craftsmen, and by all that is " put together" by art or nature. It is the " composite-
ness of any composite whole"; and, like the cognate terms KpSicns or (rvvdeais, implies
a balance or attunement. Cf. John Tate, in Class. Review, Feb. 1939.

t This general principle was clearly grasped by Mr George Rainey many years
ago, and expressed in such words as the following: "It is illogical to suppose that
in the case of vital organisms a distinct force exists to produce results perfectly
within the reach of physical agencies, especially as in many instances no end could
be attained were that the case, but that of opposing one force by another capable
of effecting exactly the same purpose." (On artificial calculi, Q.J. M.S. {Trans.
Microsc. Soc), vr, p. 49, 18.58.) Cf. also Helmholtz, infra cit. p. 9. (Mr George
Rainey, a man of learning and originality, was demonstrator of anatomy at
St Thomas's; he followed that modest calling to a great age, and is remembered
by a few old pupils with peculiar affection.)

I] OF EVOLUTION AND ENTROPY 11

world*. He may even find a certain analogy between the slow,
reluctant extension of physical laws to vital phenomena and the slow
triumphant demonstration by Tycho Brahe, Copernicus, GaHleo and
Newton (all in opposition to the Aristotelian cosmogony), that the
heavens are formed of like substance with the earth, and that the
movements of both are subject to the selfsame laws.

Organic evolution has its physical analogue in the universal law
that the world tends, in all its parts and particles, to pass from
certain less probable to certain more probable configurations or
states. This is the second law of thermodynamics. It has been
called the law of evolution of the world'f ; and we call it, after Clausius,
the Principle of Entropy, which is a literal translation of Evolution
into Greek.

The introduction of mathematical concepts into natural' science
has seemed to many men no mere stumbling-block, but a very
parting of the ways. Bichat was a man of genius, who did immense
service to philosophical anatomy, but, like Pascal, he utterly refused
to bring physics or mathematics into biology: " On calcule le retour
d'un comete, les resistances d'un fluide parcourant un canal inerte,
la Vitesse d'un projectile, etc.; mais calculer avec BorelU la force
d'un muscle, avec Keil la vitesse du sang, avec Jurine, Lavoisier et
d'autres la quantite d'air entrant dans le poumon, c'est batir sur un
sable mouvant un edifice sohde par lui-meme, mais qui tombe bientot
faute de base assureej." Comte went further still, and said that
every attempt to introduce mathematics into chemistry must be
deemed profoundly irrational, and contrary to the whole spirit of
the science §. But the great makers of modern science have all gone
the other way. Von Baer, using a bold metaphor, thought that it
might become possible " die bildenden Krafte des thierischen Korpers
. auf die allgemeinen Krafte oder Lebenserscheinun^en des Weltganzes
zuriickzufiihrenll." Thomas Young shewed, as BorelU had done,
how physics may subserve anatomy; he learned from the heart and
a,rteries that " the mechanical motions which take place in an animal's
body are regulated by the same general laws as the motions of

* So said Lobatchevsky.

t Cf. Chwolson, Lehrbuch, iii, p. 499, 1905; J. Perrin, Traitd de chimie physique,
I, p. 142, 1903; and Lotka's Elements of Physical Biology, 1925, p. 26.

t La Vie et la Mort, p. 81. § Philosophie Positive, Bk. rv.

]| Ueber Entwicklung der Thiere: Beobachtungen und Reflexionen, i, p. 22, 1828.

12 INTRODUCTORY [ch.

inanimate bodies*." And Theodore Schwann said plainly, a hun-
dred years ago, "Ich wiederhole iibrigens dass, wenn hier von einer
physikahschen Erklarung der organischen Erscheinungen die Rede
ist, darunter nicht nothwendig eine Erklarung durch die bekannten
physikalischen Krafte. . .zu verstehen ist, sondern iiberhaupt eine
Erklarung durch Krafte, die nach strengen Gesetzen der blinden
Nothwendigkeit wie die physikalischen Krafte wirken, mogen diese
Krafte auch in der anorganischen Natur auftreten oder nicht f."

Helmholtz, in a famous and influential lecture, and surely with
these very words of Schwann's in mind, laid it down as the funda-
mental principle of physiology that "there may be other agents
acting in the hving body than those agents which act in the inorganic
world ; but these forces, so far as they cause chemical and mechanical
influence in the body, must be quite of the same character as inorganic
forces : in this, at least, that their eff'ects must be ruled by necessity,
and must always be the same when acting under the same conditions ;
and so there cannot exist any arbitrary choice in the direction of their
actions." It follows further that, like the other "physical" forces,
they must be subject to mathematical analysis and deduction J.

So much for the physico-chemical problems of physiology. Apart
from these, the road of physico-mathematical or dynamical investi-
gation in morphology has found few to follow it; but the pathway
is old. The way of the old Ionian physicians, of Anaxagoras § , of
Empedocles and his disciples in the days before Aristotle, lay just
by that highway side. It was Galileo's and Borelli's way; and
Harvey's way, when he discovered the circulation of the blood ||.
It was little trodden for long afterwards, but once in a while
Swammerdam and Reaumur passed thereby. And of later years
Moseley and Meyer, Berthold, Errera and Roux have been among

* Croonian Lecture on the heart and arteries, Phil. Trans. 1809, p. 1; Collected
Works, I, p. 511.

t M ikroskopische Untersuchungen, 1839, p. 226.

J The conservation of forces applied to organic nature, Proc. Royal Inst.
April 12, 1861.

§ Whereby he incurred the reproach of Socrates, in the Phaedo. See Clerk
Maxwell on Anaxagoras as a Physicist, in Phil. Mag. (4), xlvi, pp. 453-460, 1873.

II Cf. Harvey's preface to his Exercitationes de Generatione Animalium, 1651:
"Quoniam igitur in Generatione animalium (ut etiam in caeteris rebus omnibus
de quibus aliquid scire cupimus), inquisitio omnis a caussis petenda est, praesertim
a materiali et efficiente: visum est mihi" etc.

I] OF NATURAL PHILOSOPHY 13

the little band of travellers. We need not wonder if the way be
hard to follow, and if these wayfarers have yet gathered little.
A harvest has been reaped by others, and the gleaning of the grapes
is slow.

It behoves us always to remember that in physics it has taken
great men to discover simple things. They are very great names
indeed which we couple with the explanation of the path of a stone,
the droop of a chain, the tints of a bubble, the shadows in a cup.
It is but the shghtest adumbration of a dynamical morphology that
we can hope to have until the physicist and the mathematician shall
have made these problems of ours their own, or till a new Boscovich shall
have written for the naturahst the new TheariaPhilosophiaeNaturalis.

How far even then mathematics will suffice to describe, and
physics to explain, the fabric of the body, no man can foresee. It
may be that all the laws of energy, and all the properties of matter,
and all the chemistry of all the colloids are as powerless to explain
the body as they are impotent to comprehend the soul. For my
part, I think it is not so. Of how it is that the soul informs the
body, physical science teaches me nothing; and that living matter
influences and is influenced by mind is a mystery without a clu'e.
Consciousness is not explained to my comprehension by all the
nerve-paths and neurones of the physiologist ; nor do I ask of physics
how goodness shines in one man's face, and evil betrays itself in
another. But of the construction and growth and working of the
body, as of all else that is of the earth earthy, physical science is,
in my humble opinion, our only teacher and guide.

Often and often it happens that our physical knowledge is in-
adequate to explain the mechanical working of the organism; the
phenomena are superlatively . complex, the procedure is involved
and entangled, and the investigation has occupied but a few short
lives of men. When physical science falls short of explaining the
order which reigns throughout these manifold phenomena — an order
more characteristic in its totality than any of its phenomena in
themselves — men hasten: to invoke a guiding principle, an entelechy,
or call it what you will. But all the while no physical law, any
more than gravity itself, not even among the puzzles of stereo-
chemistry or of physiological surface-action and osmosis, is known
to be transgressed by the bodily mechanism.

14 INTRODUCTORY [ch.

Some physicists declare, as Maxwell did, that atoms or molecules
more compHcated by far than the chemist's hypotheses demand, are
requisite to explain the phenomena of life. If what is impHed be
an explanation of psychical phenomena, let the point be granted at
once; we may go yet further and decHne, with Maxwell, to believe
that anything of the nature of physical complexity, however exalted,
could ever suffice. Other physicists, like Auerbach*, or Larmorj,
or Joly J, assure us that our laws of thermodynamics do not suffice,
or are inappropriate, to explain the maintenance, or (in Joly's phrase)
the accelerative absorption, of the bodily energies, the retardation
of entropy, and the long battle against the cold and darkness which
is death. With these weighty problems I am not for the moment
concerned. My sole purpose is to correlate with mathematical state-
ment and physical law certain of the simpler outward phenomena
of organic growth and structure or form, while all the while regarding
the fabric of the organism, ex hypothesi, as a material and mechanical
configuration. This is my purpose here. But I would not for the
world be thought to beheve that this is the only story which Life
and her Children have tp tell. One does not come by studying
living things for a lifetime to suppose that physics and chemistry
can account for them all§.

Physical science and philosophy stand side by side, and one
upholds the other. Without something of the strength of physics
philosophy would be weak ; and without something of philosophy's
wealth physical science would be poor. "Rien ne retirera du tissu
de la science les fils d'or que la main du philosophe y a introduits||."
But there are fields where each, for a while at least, must work alone;
and where physical science reaches its limitations physical science
itself must help us to discover. Meanwhile the appropriate and

* Ektropismus, oder die physikalische Theorie des Lebens, Leipzig, 1810.

t Wilde Lecture, Nature, March 12, 1908; ibid. Sept. 6, 1900; Aether and Matter,
p. 288. Cf. also Kelvin, Fortnightly Review, 1892, p. 313.

X The abundance of life, Proc. Roy. Dublin Soc. vii, 1890; Scientific Essays,
1915, p. 60 seq.

§ That mechanism has its share in the scheme of nature no philosopher has
denied. Aristotle (or whosoever wrote the De Mundo) goes so far as to assert that
in the most mechanical operations of nature we behold some of the divinest
attributes of God.

II J. H, Fr. Papillon, Histoire de la jyhilosophie moderne dans ses rapports avec le
developpement des sciences de la nature, i, p. 300, 1870.

I] OF LIFE ITSELF 15

legitimate postulate of the physicist, in approaching the physical
problems of the living body, is that with these physical phenomena
no ahen influence interferes. But the postulate, though it is certainly
legitimate, and though it is the proper and necessary prelude to
scientific enquiry, may some day be proven to be untrue; and its
disproof will not be to the physicist's confusion, but will come as
his reward. In dealing with forms which are so concomitant with
life that they are seemingly controlled by life, it is in no spirit of
arrogant assertiveness if the physicist begins his argument, after the
fashion of a most illustrious exemplar, with the old formula of
scholastic challenge: An Vita sit? Dico quod non.

The terms Growth and Form, which make up the title of this book,
are to be understood, as I need hardly say, in their relation to the
study of organisms. We want to see how, in some cases at least,
the forms of living things, and of the parts of living things, can be
explained by physical considerations, and to realise that in general
no organic forms exist save such as are in conformity with physical
and mathematical laws. And while growth is a somewhat vague
word for a very complex matter, which may depend on various
things, from simple imbibition of water to the complicated results
of the chemistry of nutrition, it deserves to be studied in relation
to form : whether it proceed by simple increase of size without obvious
alteration of form, or whether it so proceed as to bring about a
gradual change of form and the slow development of a more or less
complicated structure.

In the Newtonian language* of elementary physics, force is
recognised by its action in producing or in changing motion, or
in preventing change of motion or in maintaining rest. When we
deal with matter in the concrete, force does not, strictly speaking,
enter into the question, for force, unlike matter, has no independent
objective existence. It is energy in its various forms, known or
unknown, that acts upon matter. But when we abstract our
thoughts from the material to its form, or from the thing moved to
its motions, when we deal with the subjective conceptions of form,

* It is neither unnecessary nor superfluous to explain that physics is passing
through an empirical phase into a phase of pure mathematical reasoning. But
when we use physics to interpret and elucidate our biology, it is the old-fashioned
empirical physics which we endeavour, and are alone able, to apply.

16 INTRODUCTORY [ch.

or movement, or the movements that change of form impHes, then
Force is the appropriate term for our conception of the causes by
which these forms and changes of form are brought about. When
we use the term force, we use it, as the physicist always does, for
the sake of brevity, using a symbol for the magnitude and direction
of an action in reference to the symbol or diagram of a material
thing. It is a term as subjective and symbolic as form itself, and
SO' is used appropriately in connection therewith.

The form, then, of any portion of matter, whether it be living
or dead, and the changes of form which are apparent in its movements
and in its growth, may in all cases alike be described as due to
the action of force. In short, the form of an object is a "diagram
of forces," in this sense, at least, that from it we can judge of or
deduce the forces that are acting or have acted upon it: in this
strict and particular sense, it is a diagram — in the case of a sohd,
of the forces which have been impressed upon it when its conformation
was produced, together with those which enable it to retain its
conformation; in the case of a Hquid (or of a gas) of the forces which
are for the moment acting on it to restrain or balance its own
inherent mobility. In an organism, great or small, it is not merely
the nature of the motions of the hving substance which we must
interpret in terms of force (according to kinetics), but also the
conformation of the organism itself, whose permanence or equilibrium
is explained by the interaction or balance of forces, as described in
statics.

If we look at the hving cell of an Amoeba or a Spirogyra, we
see a something which exhibits certain active movements, and a
certain fluctuating, or more or less lasting, form; and its form at
a given moment, just like its motions, is to be investigated by the
help of physical methods, and explained by the invocation of the
mathematical conception of force.

Now the state, including the shape or form, of a portion of matter
is the resultant of a number of forces, which represent or symbolise
the manifestations of various kinds of energy; and it is obvious,
accordingly, that a great part of physical science must be under-
stood or taken for granted as the necessary preliminary to the
discussion on which we are engaged. But we may at least try to
indicate, very briefly, the nature of the principal forces and the

I] OF MATTER AND ENERGY 17

principal properties of matter with which our subject obhges us to
deal. Let us imagine, for instance, the case of a so-called "simple"
organism, such as Amoeba; and if our short list of its physical
•properties and conditions be helpful to our further discussion, we
need not consider how far it be complete or adequate from the
wider physical point of view*.

This portion of matter, then, is kept together by the inter-
molecular force of cohesion; in the movements of its particles
relatively to one another, and in its own movements relative to
adjacent matter, it meets with the opposing force of friction —
without the help of which its creeping movements could not be
performed. It is acted on by gravity, and this force tends (though
slightly, owing to the Amoeba's small mass, and to the small
difference between its density and that of the surrounding fluid)
to flatten it down upon the solid substance on which it may be
creeping. Our Amoeba tends, in the next place, to be deformed
by any pressure from outside, even though slight, which may be
applied to it, and this circumstance shews it to consist of matter
in a fluid, or at least semi-fluid, state: which state is further
indicated when we observe streaming or current motions in its
interior. Like other fluid bodies, its surfacef, whatsoever other
substance — gas, hquid or solid — it be in contact with, and in varying
degree according to the nature of that adjacent substance, is the
seat of molecular force exhibiting itself as a surface-tension, from
the action of which many important consequences follow, greatly
affecting the form of the fluid surface.

While the protoplasmj of the Amoeba reacts to the shghtest
pressure, and tends to "flow," and while we therefore speak of it

* With the special and impprtant properties of colloidal matter we are, for
the time being, not concerned.

t Whether an animal cell has a membrane, or only a pellicle or zona limitans,
was once deemed of great importance, and played a big part in the early contro-
versies between the cell-theory of Schwann and the protoplasma-theory of Max
Schultze and others, Dujardin came near the truth when he said, somewhat
naively, "en niant la presence d'un tegument propre, je ne pretends pas du tout
nier i'existence d'une surface."

% The word protoplasm is used here in its most general sense, as vaguely as when
Huxley spoke of it as the "physical basis of life." Its many changes and shades
of meaning in early years are discussed by Van Bambeke in the Bull. Sac. Beige
de Microscopie, xxn, pp. 1-16, 1896.

18 INTRODUCTORY [ch.

as a fluid*, it is evidently far less mobile than such a fluid (for
instance) as water, but is rather Uke treacle in its slow creeping
movements as it changes its shape in response to force. Such fluids
are said to have a high viscosity, and this viscosity obviously acts
in the way of resisting change of form, or in other words of
retarding the efl'ects of any disturbing action of force. When the
viscous fluid is capable of being drawn out into fine threads, a
property in which we know that some Amoebae differ greatly from
others, we say that the fluid is also viscid, or exhibits viscidity.
Again, not by virtue of our Amoeba being liquid, but at the same
time in vastly greater measure than if it were a sohd (though far less
rapidly than if it were a gas), a process of molecular diffusion is
constantly going on within its substance, by which its particles
interchange their places within the mass, while surrounding fluids,
gases and soUds in solution diffuse into and out of it. In so far
as the outer wall of the cell is different in character from the
interior, whether it be a mere pelhcle as in Amoeba or a firm
cell-wall as in Protococcus, the diffusion which takes place throtigh
this wall is sometimes distinguished under the term osmosis.

Within the cell, chemical forces are at work, and so also in all
probabihty (to judge by analogy) are electrical forces; and the
organism reacts also to forces from without, that have their origin
in chemical, electrical and thermal influences. The processes of
diffusion and of chemical activity within the cell result, by the
drawing in of water, salts, and food-material with or without
chemical transformation into protoplasm, in growth, and this com-
plex phenomenon we shall usually, without discussing its nature
and origin, describe and picture as a force. Indeed we shall
manifestly be incHned to use the term growth in two senses, just
indeed as we do in the case of attraction or gravitation, on the one
hand as a process, and on the other as a force.

In the phenomena of cell-division, in the attractions or repulsions
of the parts of the dividing nucleus, and in the " caryokinetic "
figures which appear in connection with it, we seem to see in
operation forces and the effects of forces which have, to say the

* One of the first statements which Dujardin made about protoplasm (or, as
he called it, sarcode) was that it was not a fluid; and he relied greatly on this fact
to shew that it was a living, or an organised, structure.

I] OF VITAL PHENOMENA 19

least of it, a close analogy with known physical phenomena : and
to this matter we shall presently return. But though they resemble
known physical phenomena, their nature is still the subject of much
dubiety and discussion, and neither the forms produced nor the
forces at work can yet be satisfactorily and simply explained. We
may readily admit then, that, besides phenomena which are obviously
physical in their nature, there are actions visible as well as invisible
taking place within living cells which our knowledge does not permit
us to ascribe with certainty to any known physical force; and it
may or may not be that these phenomena will yield in time to the
methods of physical investigation. Whether they do or no, it is
plain that we have no clear rule or guidance as to what is "vital"
and what is not; the whole assemblage of so-called vital phenomena,
or properties of the organism, cannot be clearly classified into those
that are physical in origin and those that are sui generis and peculiar
to living things. All we can do meanwhile is to analyse, bit by bit,
those parts of the whole to which the ordinary laws of the physical
forces more or less obviously and clearly and indubitably apply.

But even the ordinary laws of the physical forces are by no means
simple and plain. In the winding up of a clock (so Kelvin once
said), and in the properties of matter which it involves, there is
enough and more than enough of mystery for our limited under-
standing: "a watchspring is much farther beyond our understanding
than a gaseous nebula." We learn and learn, but never know all,
about the smallest, humblest thing. So said St Bonaventure : " Si per
multos annos viveres, adhuc naturam unius festucae seu muscae seu
minimae creaturae de mundo ad plenum cognoscere non valeres*."
There is a certain fascination in such ignorance ; and we learn (like
the Abbe Galiani) without discouragement that Science is "plutot
destine a etudier qu'a connaitre, a chercher qu'a trouver la verite."

Morphology is not only a study of material things and of the forms
of material things, but has its dynamical aspect, under which we
deal with the interpretation, in term^ of force, of the operations of
Energyf . And here it is well worth while to remark that, in deahng

* Op. V, p. 541 ; cit. E. Gilson.

t This is a great theme. Boltzmann, writing in 1886 on the second law of
thermodynamics, declared that available energy was the main object at stake
in the struggle for existence and the evolution of the world. Cf. Lotka, The
energetics of evolution, Proc. Nat. Acad. Sci. 1922, p. 147.

20 INTRODUCTORY [ch.

with the facts of embryology or the phenomena of inheritance, the
common language of the, books seems to deal too much with the
material elements concerned, as the causes of development, of
variation or of hereditary transmission. Matter as such produces
nothing, changes nothing, does nothing; and however convenient
it may afterwards be to abbreviate our nomenclature and our
descriptions, we must most carefully realise in the outset that the
spermatozoon, the nucleus, the chromosomes or the germ-plasma
can never act as matter alone, but only as seats of energy and as
centres of force. And this is but an adaptation (in the light, or
rather in the conventional symboHsm, of modern science) of the old
saying of the philosopher : apx^ yo.p r) <j>voLs fxdXXov rrjs vXrjg.

Since this book was written, some five and twenty years ago,
certain great physico-mathematical concepts have greatly changed.
Newtonian mechanics and Newtonian concepts of space and time
are found unsuitable, even untenable or invahd, for the all but
infinitely great and the all but infinitely small. The very idea of
physical causation is said to be illusory, and the physics of the
atom and the electron, and of the quantum theory, are to be
elucidated by the laws of probability rather than by the concept
of causation and its effects. But the orders of magnitude, whether
of space or time, within which these new concepts become useful,
or hold true, lie far away. We distinguish, and can never help
distinguishing, between the things which are of our own scale and
order, to which our minds are accustomed and our senses attuned,
and those remote phenomena which ordinary standards fail to
measure, in regions where (as Robert Louis Stevenson said) there
is no habitable city for the mind of man.

It is no wonder if new methods, new laws, new words, new modes
of thought are needed when we make bold to contemplate a Universe
within which all Newton's is but a speck. But the world of the
Hving, wide as it may be, is bounded by a famihar horizon within
which our thoughts and senses are at home, our scales of time and
magnitude suffice, and the Natural Philosophy of Newton and
Gahleo rests secure.

We start, like Aristotle, with our own stock-in-trade of know-
ledge: dpKTeov OLTTO Tcov rjfjuv yvajpLjjLojv. And only when we are

I] OF NEWTONIAN PHYSICS 21

steeped to the marrow (as Henri Poincare once said) in the old laws,
and in no danger of forgetting them, may we be allowed to learn
how they have their remote but subtle limitations, and cease afar
off to be more than approximately true *. Kant's axiom of causahty,
that it is denknotwendig — indispensable for thought — remains true
however physical science may change. His later aphorism, that all
changes take place subject to the law which links cause and effect
together — "alle Veranderungen geschehen nach dem Gesetz der
Verkniipfung von Ursache und Wirkung" — is still an axiom a priori,
independent of experience: for experience itself depends upon its
truth t-

* So Max Planck himself says somewhere: "In my opinion the teaching of
mechanics will still have to begin with Newtonian force, just as optics begins in
the sensation of colour and thermodynamics with the sensation of warmth,
despite th^ fact that a more precise basis is substituted later on."

t "Weil er [der Grundsatz das Kausalverhaltnisses] selbst der grund der Moglich-
keit einer solchen Erfahrung ist": Kritik d. reinen Vernunft, ed. Odicke, 1889, p. 221.
Cf. also G. W. Kellner, Die Kausalitat in der Physik, Ztschr.f. Physik, lx,iv, pp. 568-
580. 1930.

CHAPTER II

ON MAGNITUDE

To terms of magnitude, and of direction, must we refer all our
conceptions of Form. For the form of an object is defined when we
know its magnitude, actual or relative, in various directions; and
Growth involves the same concepts of magnitude and direction,
related to the further concept, or "dimension," of Time. Before
we proceed to the consideration of specific form, it will be well to
consider certain general phenomena of spatial magnitude, or of the
extension of a body in the several dimensions of space.

We are taught by elementary mathematics — and by Archimedes
himself — that in similar figures the surface increases as the square,
and the volume as the cube, of the linear dimensions. If we take
the simple case of a sphere, with radius r, the area of its surface is
equal to 4:7rr^, and its volume to ^ttt^ ; from which it follows that the
ratio of its volume to surface, or V/S, is Jr. That is to say, VfS
varies as r; or, in other words, the larger the sphere by so much the
greater will be its volume (or its mass, if it be uniformly dense
throughout) in comparison with its superficial area. And, taking
L to represent any linear dimension, we may write the general
equations in the form

Soz L\ F oc L^

or iS = kL\ and V = k'L\

where k, k', are "factors of proportion,"

V V k

and ^ cc L, or — = j-, L = KL.

o ok

So, in Lilliput, "His Majesty's Ministers, finding that Gulhver's
stature exceeded theirs in the proportion of twelve to one, concluded
from the similarity of their bodies that his must contain at least
1728 [or 12^] of theirs, and must needs be rationed accordingly*."

* Likewise Gulliver had a whole Lilliputian hogshead for his half-pint of wine:
in the due proportion of 1728 half-pints, or 108 gallons, equal to one pipe or

CH. II] OF DIMENSIONS 23

From these elementary principles a great many consequences
follow, all more or less interesting, and some of them of great
importance. In the first place, though growth in length (let us say)
and growth in volume (which is usually tantamount to mass or
weight) are parts of one and the same process or phenomenon, the
one attracts our attention by its increase very much more than the
other. For instance a fish, in doubhng its length, multiphes its
weight no less than eight times; and it all but doubles its weight in
growing from four inches long to five.

In the second place, we see that an understanding of the correla-
tion between length and weight in any particular species of animal,
in other words a determination of k in the formula W = k.L^,
enables us at any time to translate the one magnitude into the other,
and (so to speak) to weigh the animal with a measuring-rod; this,
however, being always subject to the condition that the animal shall
in no way have altered its form, nor its specific gravity. That its
specific gravity or density should materially or rapidly alter is not
very likely; but as long as growth lasts changes of form, even
though inappreciable to the eye, are apt and hkely to occur. Now
weighing is a far easier and far more accurate operation than
measuring; and the measurements which would reveal slight and
otherwise imperceptible changes in the form of a fish — slight relative
differences between length, breadth and depth, for instance — would
need to be very dehcate indeed. But if we can make fairly accurate
determinations of the length, which is much the easiest linear
dimension to measure, and correlate it with the weight, then the
value of k, whether it varies or remains constant, will tell us at once
whether there has or has not been a tendency to alteration in the
general form, or, in other words, a difference in the rates of growth
in different directions. To this subject we shall return, when we
come to consider more particularly the phenomenon of rate of growth.

double-hogshead. But Gilbert White of Selborne could not see what was plain
to the Lilliputians; for finding that a certain little long-legged bird, the stilt,
weighed 4J oz. and had legs 8 in. long, he thought that a flamingo, weighing 4 lbs.,
should have legs 10 ft. long, to be in the same proportion as the stilt's. But
it is obvious to us that, as the weights of the two birds are as 1 : 15, so the legs
(or other linear dimensions) should be as the cube-roots of these numbers, or
nearly as 1 : 2^. And on this scale the flamingo's legs should be, as they actually
are, about 20 in. long.

24 ON MAGNITUDE [ch.

We are accustomed to think of magnitude as a purely relative
matter. We call a thing big or little with reference to what it is
wont to be, as when we speak of a small elephant or a large rat ; and
we are apt accordingly to suppose that size makes no other or more
essential difference, and that Lilliput and Brobdingnag* are all
alike, according as we look at them through one end of the glass
or the other. Gulliver himself declared, in Brobdingnag, that
"undoubtedly philosophers are in the right when they tell us that
nothing is great and little otherwise than by comparison": and
Oliver Heaviside used to say, in like manner, that there is no
absolute scale of size in the Universe, for it is boundless towards
the great and also boundless towards the small. It is of the very
essence of the Newtonian philosophy that we should be able to
extend our concepts and deductions from the one extreme of magni-
tude to the other; and Sir John Herschel said that "the student
must lay his account to finding the distinction of great and little
altogether annihilated in nature."

All this is true of number, and of relative magnitude. The Universe
has its endless gamut of great and small, of near and far, of many
and few. Nevertheless, in physical science the scale of absolute
magnitude becomes a very real and important thing; and a new
and deeper interest arises out of the changing ratio of dimensions
when we come to consider the inevitable changes of physical rela-
tions with which it is bound up. The effect of scale depends not on
a thing in itself, but in relation to its whole environment or milieu ;
it is in conformity with the thing's "place in Nature," its field of
action and reaction in the Universe. Everywhere Nature works
true to scale, and everything has its proper size accordingly. Men
and trees, birds and fishes, stars and star-systems, have their
appropriate dimensions, and their more or less narrow range of
absolute magnitudes. The scale of human observation and ex-
perience lies within the narrow bounds of inches, feet or miles, all
measured in terms drawn from our own selves or our own doings.
Scales which include light-years, parsecs. Angstrom units, or atomic

* Swift paid close attention to the arithmetic of magnitude, but none to its
physical aspect. See De Morgan, on Lilliput, in N. and Q. (2), vi, pp. 123-125,
1858. On relative magnitude see also Berkeley, in his Essay towards a New Theory
of Visio7i, 1709.

II] THE EFFECT OF SCALE 25

and sub-atomic magnitudes, belong to other orders of things and
other principles of cognition.

A common effect of scale is due to the fact that, of the physical
forces, some act either directly at the surface of a body, or otherwise
in proportion to its surface or area; while others, and above all
gravity, act on all particles, internal and external alike, and exert
a force which is proportional to the mass, and so usually to the
volume of the, body.

A simple case is that of two similar weights hung by two similar
wires. The forces exerted by the weights are proportional to their
masses, and these to their volumes, and so to the cubes of the
several Hnear dimensions, including the diameters of the wires.
But the areas of cross-section of the wires are as the squares of the
said linear dimensions; therefore the stresses in the wires 'per unit
area are not identical, but increase in the ratio of the linear dimen-
sions, and the larger the structure the more severe the strain becomes :

Force l^

A^ ^ r^ ^ ^'

and the less the wires are capable of supporting it.

In short, it often happens that of the forces in action in a system
some vary as one power and some as another, of the masses, distances
or other magnitudes involved; the "dimensions" remain the same
in our equations of equilibrium, but the relative values alter with
the scale. This is known as the "Principle of Similitude," or of
dynamical similarity, and it and its consequences are of great
importance. In a handful of matter cohesion, capillarity, chemical
affinity, electric charge are all potent; across the solar system
gravitation* rules supreme; in the mysterious region of the nebulae,
it may haply be that gravitation grows negligible again.

To come back to homelier things, the strength of an iron girder
obviously varies with the cross-section of its members, and each
cross-section varies as the square of a linear dimension; but the
weight of the whole structure varies as the cube of its linear dimen-

* In the early days of the theory of gravitation, it was deemed especially
remarkable that the action of gravity "is proportional to the quantity of solid
matter in bodies, and not to their surfaces as is usual in mechanical causes; this
power, therefore, seems to surpass mere mechanism" (Colin Maclaurin, on Sir
Isaac Newton's Philosophical Discoveries, iv, 9).

26 ON MAGNITUDE [ch.

sions. It follows at once that, if we build two bridges geometrically
similar, the larger is the weaker of the two*, and is so in the ratio
of their linear dimensions. It was elementary engineering experience
such as this that led Herbert Spencer to apply the principle of
simihtude to biologyf.

But here, before w^e go further, let us take careful note that
increased weakness is no necessary concomitant of increasing size.
There are exceptions to the rule, in those exceptional cases where we
have to deal only with forces which vary merely with the area on
which they impinge. - If in a big and a httle ship two similar masts
carry two similar sails, the two sails will be similarly strained, and
equally stressed at homologous places, and alike suitable for resisting
the force of the same wind. Two similar umbrellas, however
differing in size, will serve ahke in the same weather; and the
expanse (though not the leverage) of a bird's wing may be enlarged
with little alteration.

The principle of similitude had been admirably apphed in a few
clear instances by Lesage J, a celebrated eighteenth-century physician,
in an unfinished and unpublished work. Lesage argued, for example,
that the larger ratio of surface to mass in a small animal would lead
to excessive transpiration, were the skin as "porous" as our own;
and that we may thus account for the hardened or thickened skins
of insects and many other small terrestrial animals. Again, since
the weight of a fruit increases as the cube of its linear dimensions,
while the strength of the stalk increases as the square, it follows
that the stalk must needs grow out of apparent due proportion to
the fruit: or, alternatively, that tall trees should not bear large

* The subject is treated from the engineer's point of view by Prof. James
Thomson, Comparison of similar structures as to elasticity, strength and stability,
Coll. Papers, 1912, pp. 361-372, and Trans. Inst. Engineers, Scotland, 1876; also
by Prof. A. Barr, ibid. 1899. See also Rayleigh, Nature, April 22, 1915; Sir G.
Greenhill, On mechanical similitude, Math. Gaz. March 1916, Coll. Works, vi,
p. 300. For a mathematical account, sec (e.g.) P. VV. Bridgeman, Dimensional
Analysis (2nd ed.), 1931, or F. W. Lanchester, The Theory of Dimensions, 1936.

t Herbert Spencer, The form of the earth, etc., Phil. Mag. xxx, pp. 194-6,
1847; also Principles of Biology, pt. ii, p. 123 seq., 1864.

I See Pierre Prevost, Notices de la vie et des ecrits de Lesage, 1805. George
Louis Lesage, born at Geneva in 1724, devoted sixty-three years of a life of eighty
to a mechanical theory of gravitation; see W. Thomson (Lord Kelvin), On the
ultramundane corpuscles of Lesage, Proc. E.S.E. vii, pp. 577-589, 1872; Phil. Mag.
XLV, pp. 321-345, 1873; and Clerk Maxwell, art. "Atom," Encyd. Brit. (9), p.' 46.

II] THE PRINCIPLE OF SIMILITUDE 27

fruit on slender branches, and that melons and pumpkins must lie
upon the ground. And yet again, that in quadrupeds a large head
must be supported on a neck which is either excessively thick and
strong like a bull's, or very short like an elephant's*.

But it was Gahleo who, wellnigh three hundred years ago, had
first laid down this general principle of simiUtude; and he did so
with the utmost possible clearness, and with a great wealth of illustra-
tion drawn from structures living and deadf. He said that if we
tried building ships, palaces or temples of enormous size, yards,
beams and bolts would cease to hold together; nor can Nature
grow a tree nor construct an animal beyond a certain size, while
retaining the proportions and employing the materials which suffice
in the case of a smaller structure J. The thing will fall to pieces of
its own weight unless we either change its relative proportions, which
will at length cause it to become clumsy, monstrous and inefficient,
or else we must find new material, harder and stronger than was
used before. Both processes are famihar to us in Nature and in
art, and practical apphcations, undreamed of by Gahleo, meet us at
every turn in this modern age of cement and steel §.

Again, as Galileo was also careful to explain, besides the questions
of pure stress and strain, of the strength of muscles to hft an
increasing weight or of bones to resist its crushing stress, we have
the important question of bending ynornents. This enters, more or
less, into our whole range of problems ; it aifects the whole form of
the skeleton, and sets a limit to the height of a tall tree||.

* Cf. W. Walton, On the debility of large animals and trees, Quart. Journ.
of Math. IX, pp. 179-184, 1868; also L. J. Henderson, On volume in Biology,
Proc. Amer. Acad. Sci. ii, pp. 654-658, 1916; etc.

t Discorsi e Dimostrazioni matematiche, intorno a due nuove scienze attenenti
alia Mecanica ed ai Muovimenti Locali: appresso gli Elzevirii, 1638; Opere,
ed. Favaro, viir, p. 169 seq. Transl. by Henry Crew and A. de Salvio, 1914, p. 130.

X So Werner remarked that Michael Angelo and Bramanti could not have built
of gypsum at Paris on the scale they built of travertin at Rome.

§ The Chrysler and Empire State Buildings, the latter 1048 ft. high to the foot
of its 200 ft. "mooring mast," are the last word, at present, in this brobdingnagian
architecture.

II It was Euler and Lagrange who first shewed (about 1776-1778) that a column
of a certain height would merely be compressed, but one of a greater height would
be bent by its own weight. See Euler, De altitudine columnarum etc.. Acta Acad.
Sci. Imp. Petropol. 1778, pp. 163-193; G, Greenhill, Determination of the greatest
height to which a tree of given proportions can grow, Cambr. Phil. Soc. Proc. rv,
p. 65, 1881, and Chree, ibid, vu, 1892.

28 ON MAGNITUDE [ch.

We learn in elementary mechanics the simple case of two similar
beams, supported at both ends and carrying no other weight than
their own. Within the limits of their elasticity they tend to be
deflected, or to sag downwards, in proportion to the squares of their
linear dimensions ; if a match-stick be two inches long and a similar
beam six feet (or 36 times as long), the latter will sag under its own
weight thirteen hundred times as much as the other. To counteract
this tendency, as the size of an animal increases, the limbs tend to
become thicker and shorter and the whole skeleton bulkier and
heavier; bones make up some 8 per cent, of the body of mouse or wren,
13 or 14 per cent, of goose or dog, and 17 or 18 per cent, of the body
of a man. Elephant and hippopotamus have grown clumsy as well as
big, and the elk is of necessity less graceful than the gazelle. It is of
high interest, on the other hand, to observe how little the skeletal
proportions differ in a httle porpoise and a great whale, even in the
limbs and hmb-bones ; for the whole influence of gravity has become
neghgible, or nearly so, in both of these.

In ifhe problem of the tall tree we have to determine the point
at which the tree will begin to bend under its own weight if it be
ever so little displaced from the perpendicular*. In such an
investigation we have to make certain assumptions — for instance
that the trunk tapers uniformly, and that the sectional area of the
branches varies according to some definite law, or (as Ruskin
assumed) tends to be constant in any horizontal plane; and the
mathematical treatment is apt to be somewhat difficult. But
Greenhill shewed, on such assumptions as the above, that a certain
British Columbian pine-tree, of which the Kew flag-staff, which is
221 ft. high and 21 inches in diameter at the base, was made, could
not possibly, by theory, have grown to more than about 300 ft. It
is very curious that Galileo had suggested precisely the same height
(ducento braccie alta) as the utmost limit of the altitude of a tree.
In general, as Greenhill shewed, the diameter of a tall homogeneous
body must increase as the power 3/2 of its height, which accounts
for the slender proportions of young trees compared with the squat

* In like manner the wheat-straw bends over under the weight of the loaded
ear, and the cat's tail bends over when held erect — not because, they "possess
flexibility," but because they outstrip the dimensions within which stable equi-
librium is possible in a vertical position. The kitten's tail, on the other hand,
stands up spiky and straight.

II] OF THE HEIGHT OF A TREE 29

or stunted appearance of old and large ones*. In short, as Goethe
says in Dicktung und Wahrheit, "Es ist dafiir gesorgt dass die Baume
nicht in den Himmel wachsen."

But the tapering pine-tree is but a special case of a wider problem.
The oak does not grow so tall as the pine-tree, but it carries a heavier
load, and its boll, broad-based upon its spreading roots, shews a
different contour. Smeaton took it for the pattern of his Hghthouse,
and Eiffel built his great tree of steel, a thousand feet high, to a
similar but a stricter plan. Here the profile of tower or tree follows,
or tends to follow, a logarithmic curve, giving equal strength
throughout, according to a principle which we shall have occasion
to discuss later on, when we come to treat of form and mechanical
efficiency in the skeletons of animals. In the tree, moreover,
anchoring roots form powerful wind-struts, and are most de-
veloped opposite to the direction of the prevailing winds; for the
lifetime of a tree is affected by the frequency of storms, and its
strength is related to the wind-pressure which it must needs with-
standf.

Among animals we see, without the help of mathematics or of
physics, how small birds and beasts are quick and agile, how slower
and sedater movements come with larger size, and how exaggerated
bulk brings with it a certain clumsiness, a certain inefficiency, an
element of risk and hazard, a preponderance of disadvantage. The
case was well put by Owen, in a passage which has an interest of
its own as a premonition, somewhat Hke De. Candolle's, of the
"struggle for existence." Owen wrote as follows J: " In proportion
to the bulk of a species is the difficulty of the contest which, as a
living organised whole, the individual of each species has to maintain
against the surrounding agencies that are ever tending to dissolve
the vital bond, and subjugate the Hving matter to the ordinary
chemical and physical forces. Any changes, therefore, in such
external conditions as a species may have been original|y adapted

* The stem of the giant bamboo may attain a height of 60 metres while not more
than about 40 cm. in diameter near its base, which dimensions fall not far short
of the theoretical limits; A. J. Ewart, Phil. Trans, cxcviii, p. 71, 1906.

t Cf. {int. al.) T. Fetch, On buttress tree-roots, Ann. R. Bot. Garden, Peradenyia,
XI, pp. 277-285, 1930. Also au interesting paper by James Macdonald, on The
form of coniferous trees. Forestry, vi, 1 and 2, 1931/2.

X Trans. Zool. Soc. iv, p. 27, 1850.

30 ON MAGNITUDE [ch.

to exist in, will militate against that existence in a degree
proportionate, perhaps in a geometrical ratio, to the bulk of the
species. If a dry season be greatly prolonged, the large mammal
will suffer from the drought sooner than the small one; if any
alteration of climate affect the quantity of vegetable food, the
bulky Herbivore will be the first to feel the effects of stinted
nourishment."

But the principle of GaHleo carries us further and along more
certain lines. The strength of a muscle, like that of a rope or
girder, varies with its cross-section; and the resistance of a bone
to a crushing stress varies, again hke our girder, with its cross-
section. But in a terrestrial animal the weight which tends to
crush its limbs, or which its muscles have to move, varies as the
cube of its hnear dimensions; and so, to the possible magnitude
of an animal, living under the direct action of gravity, there is a
definite limit set. The elephant, in the dimensions of its limb-bones,
is already shewing signs of a tendency to disproportionate thickness
as compared with the smaller mammals; its movements are in
many ways hampered and its agility diminished: it is already
tending towards the maximal Hmit of size which the physical forces
permit*. The spindleshanks of gnat or daddy-long-legs have their
own factor of safety, conditional on the creature's exiguous bulk
and weight; for after their own fashion even these small creatures
tend towards an inevitable limitation of their natural size. But, as
Gahleo also saw, if the animal be wholly immersed in water like the
whale, or if it be partly so, as was probably the case with the giant
reptiles of the mesozoic age, then the weight is counterpoised to
the extent of an equivalent volume of water, and is completely
counterpoised if the density of the animal's body, with the included
air, be identical (as a whale's very nearly is) with that of the water
around^. Under these circumstances there is no longer the same
physical barrier to the indefinite growth of the animal. Indeed, in the
case of the aquatic animal, there is, as Herbert Spencer pointed out,

* Cf. A. Rauber, Galileo iiber Knochenformen, Morphol. Jahrb. vii, p. 327, 1882.

t Cf. W. S. Wall, A New Sperm Whale etc., Sydney, 1851, p. 64: "As for
the immense size of Cetacea, it evidently proceeds from their buoyancy in the
medium in which they live, and their being enabled thus to counteract the force of

gravity."

II] OF THE SPEED OF A SHIP 31

a distinct advantage, in that the larger it grows the greater is its
speed. For its available energy depends on the mass of its muscles,
while its motion through the water is opposed, not by gravity, but
by "skin-friction," which increases only as the square of the linear
dimensions*: whence, other things being equal, the bigger the ship
or the bigger the fish the faster it tends to go, but only in the ratio
of the square root of the increasing length. For the velocity (F)
which the fish attains depends on the work (W) it can do and the
resistance (R) it must overcome. Now we^ have seen that the
dimensions of W are l^, and of R are l'^ ; and by elementary mechanics

WocRV^ or F^oc^.

73

Therefore V^oCr-=l, and F oc Vl.

This is what is known as Fronde's Law, of the correspondence
of speeds — a simple and most elegant instance of "dimensional
theory!."

But there is often another side to these questions, which makes
them too complicated to answer in a word. For instance, the work
(per stroke) of which two similar engines are capable should vary as
the cubes of their linear dimensions, for it varies on the one hand
with the area of the piston, and on the other with the length of the
stroke; so is it likewise in the animal, where the corresponding
ratio depends on the cross-section of the muscle, and on the distance
through which it contracts. But in two similar engines, the available
horse-power varies as the square of the linear dimensions, and not
as the cube ; and this for the reason that the actual energy developed
depends on the heating-surface of the boiler J. So likewise must

* We are neglecting "drag" or "head-resistance," which, increasing as the cube
of the speed, is a formidable obstacle to an unstreamlined body. But the perfect
streamlining of whale or fish or bird lets the surrounding air or water behave like
a perfect fluid, gives rise to no "surface of discontinuity," and the creature passes
through it without recoil or turbulence. Froude reckoned skin-friction, or surface-
resistance, as equal to that of a plane as long as the vessel's water-line, and of area
equal to that of the wetted surface of the vessel.

t Though, as Lanchester says, the great designer "was not hampered by a
knowledge of the theory of dimensions."

X The analogy is not a very strict or complete one. We are not taking account,
for instance, of the thickness of the boiler-plates.

32 ON MAGNITUDE [ch.

there be a similar tendency among animals for the rate of supply
o^ kinetic energy to vary with the surface of the lung, that is to say
(other things being equal) with the square of the linear dimensions
of the animal ; which means that, caeteris paribus, the small animal
is stronger (having more power per unit weight) than a large one.
We may of course (departing from the condition of similarity) increase
the heating-surface of the boiler, by means of an internal system of
tubes, without increasing its outward dimensions, and in this very
way Nature increases the respiratory surface of a lung by a complex
system of branching tubes and minute air-cells ; but nevertheless in
two similar and closely related animals, as also in two steam-engines
of the same make, the law is bound to hold that the rate of working
tends to vary with the square of the linear dimensions, according to
Froude's law of steamship comparison. In the case of a very large
ship, built for speed, the difficulty is got over by increasing the size
and number of the boilers, till the ratio between boiler-room and
engine-room is far beyond what is required in an ordinary small
vessel*; but though we find lung-space increased among animals
where greater rate of working is required, as in general among birds,
I do not know that it can be shewn to increase, as in the "over-
boilered" ship, with the size of the animal, and in a ratio which
outstrips that of the other bodily dimensions. If it be the case then,
that the working mechanism of the muscles should be able to exert
a force proportionate to the cube of the linear bodily dimensions,

* Let L be the length, S the (wetted) surface, T the tonnage, D the displacement
(or volume) of a ship; and let it cross the Atlantic at a speed V. Then, in com-
paring two ships, similarly constructed but of different magnitudes, we know that
L=V\ S=L^ = V\ D = T = L^=V^; also B (resistance) =/Sf. F^^ F«; H (horse-
power) = i2 . F = F' ; and the coal (C) necessary for the voy a,ge— HjV = V^. That
is to say, in ordinary engineering language, to increase the speed across the Atlantic
by 1 per cent, the ship's length must be increased 2 per cent., her tonnage or
displacement 6 per cent., her coal- consumption also 6 per cent., her horse-power,
and therefore her boiler-capacity, 7 per cent. Her bunkers, accordingly, keep
pace with the enlargement of the ship, but her boilers tend to increase out of
proportion to the space available. Suppose a steamer 400 ft. long, of 2000 tons,
2000 H.P., and a speed of 14 knots. The corresponding vessel of 800 ft. long should
develop a speed of 20 knots (I : 2 :: 14^ : 20^), her tonnage would be 16,000, her
H.p. 25,000 or thereby. Such a vessel would probably be driven by four propellers
instead of one, each carrying 8000 h.p. See (int. al.) W. J. Millar, On the most
economical speed to drive a steamer, Proc. Edin. Math. Soc. vii, pp. 27-29, 1889;
Sir James R. Napier, On the most profitable speed for a fully laden cargo steamer
for a given voyage, Proc. Phil. Soc, Glasgow, vi, pp. 33-38, 1865.

II] OF FROUDE'S LAW 33

while the respiratory mechanism can only supply a store of energy
at a rate proportional to the square of the said dimensions, the
singular result ought to follow that, in swimming for instance, the
larger fish ought to be able to put on a spurt of speed far in excess
of the smaller one; but the distance travelled by the year's end
should be very much ahke for both of them. And it should also
follow that the curve of fatigue is a steeper one, and the staying
power less, in the smaller than ixx the larger individual. This is the
c^ase in long-distance racing, where neither draws far ahead until
the big winner puts on his big spurt at the end; on which is based
an aphorism of the turf, that "a good big 'un is better than a good
httle 'un." For an analogous reason wise men know that in the
'Varsity boat-race it is prudent and judicious to bet on the heavier
crew.

Consider again the dynamical problem of the movements of the
body and the hmbs. The work done (W) in moving a Hmb, whose
weight is p, over a distance s, is measured hy ps; p varies as the
cube of the hnear dimensions, and s, in ordinary locomotion, varies
as the linear dimensions, that is to say as the length of Hmb :

Wocpsocl^ xl^lK

But the work done is limited by the power available, and this
varies as the mass of the muscles, or as l^; and under this hmitation
neither p nor s increase as they would otherwise tend to do. The
limbs grow shorter, relatively, as the animal grows bigger; and
spiders, daddy-long-legs and such-hke long-limbed creatures attain
no great size.

Let us consider more closely the actual energies of the body.
A hundred years ago, in Strasburg, a physiologist and a mathema-
tician were studying the temperature of warm-blooded animals*.
The heat lost must, they said, be proportional to the surface of the
animal : and the gain must be equal to the loss, since the temperature
of the body keeps constant. It would seem, therefore, that the
heat lost by radiation and that gained by oxidation vary both alike,
as the surface-area, or the square of the Hnear dimensions, of the
animal. But this result is paradoxical; for whereas the heat lost

* MM. Rameaux et Sarrus, Bull. Acad. R. de Me'decine, in, pp. 1094-1100,
1838-39.

34 ON MAGNITUDE [ch.

may well vary as the surface-area, that produced by oxidation
ought rather to vary as the bulk of the animal: one should vary
as the square and the other as the cube of the Hnear dimensions.
Therefore the ratio of loss to gain, hke that of surface to volume,
ought to increase as the size of the creature diminishes. Another
physiologist, Carl Bergmann*, took the case a step further. It was he,
by the way, whp first said that the real distinction was not between
warm-blooded and cold-blooded animals, but between those of
constant and those of variable temperature: and who coined the
terms homoeothermic and poecilothermic which we use today. He
was driven to the conclusion that the smaller animal does produce
more heat (per unit of mass) than the large one, in order to keep
pace with surface-loss; and that this extra heat-production means
more energy spent, more food consumed, more work donef. Sim-
plified as it thus was, the problem still perplexed the physiologists
for years after. The tissues of one mammal are much like those of
another. We can hardly imagine the muscles of a small mammal
to produce more heat {caeteris paribus) than those of a large ; and
we begin to wonder whether it be not nervous excitation, rather than
quahty of muscular tissue, which determines the rate of oxidation
and the output of heat. It is evident in certain cases, and may be
a general rule, that the smaller animals have the bigger brains;
"plus I'animal est petit," says M. Charles Richet, "plus il a des
echanges chimiques actifs, et plus son cerveau est volumineuxj."
That the smaller animal needs more food is. certain and obvious.
The amount of food and oxygen consumed by a small flying insect
is enormous; and bees and flies and hawkmoths and humming-

* Carl Bergmann, Verhaltnisse der Warmeokonomie der Tiere zu ihrer Grosse,
Gottinger Studien, i, pp. 594-708, 1847 — a very original paper.

t The metabolic activity of sundry mammals, per 24 hours, has been estimated
as follows :

Weight (kilo.) Calories per kilo.
Guinea-pig 0-7 223

Rabbit 2 58

Man 70 33

Horse 600 22

Elephant 4000 13

Whale 150000 circa 1-7

J Ch. Richet, Recherches de calorimetrie. Arch, de Physiologie (3), vi, pp. 237-291,
450-497, 1885. Cf. also an interesting historical account by M. Elie le Breton,
Sur la notion de "masse protoplasmique active": i. Probl^mes poses par la
signification de la loi des surfaces, ibid. 1906, p. 606.

II] OF BERGMANN'S LAW 35

birds live on nectar, the richest and most concentrated of foods*.
Man consumes a fiftieth part of his own weight of food daily, but
a mouse will eat half its own weight in a day; its rate of Hving is
faster, it breeds faster, and old age comes to it much sooner than
to man. A warm-blooded animal much smaller than a mouse
becomes an impossibihty; it could neither obtain nor yet digest the
food required to maintain its constant temperature, and hence no
mammals and no birds are as small as the smallest frogs or fishes.
The disadvantage of small size is all the greater when loss of heat
is accelerated by conduction as in the Arctic, or by convection as
in the sea. The far north is a home of large birds but not of small ;
bears but not mice five through an Arctic winter; the 'least of the
dolphins Hve in warm waters, and there are no small mammals in
the sea. This principle is sometimes spoken of as Bergmann's Law.

The whole subject of the conservation of heat and the maintenance
of an all but constant temperature in warm-blooded animals interests
the physicist and the physiologist ahke. It drew Kelvin's attention
many years agof, and led him to shew, in a curious paper, how
larger bodies are kept warm by clothing while smaller are only
cooled the more. If a current be passed through a thin wire, of
which part is covered and part is bare, the thin bare part may glow
with heat, while convection-currents streaming round the covered
part cool it off and leave it in darkness. The hairy coat of very
small animals is apt to look thin and meagre, but it may serve them
better than a shaggier covering.

Leaving aside the question of the supply of energy, and keeping
to that of the mechanical efficiency of the machine, we may find
endless biological illustrations of the principle of simihtude. All
through the physiology of locomotion we meet with it in various
ways : as, for instance, when we see a cockchafer carry a plate many
times its own weight upon its back, or a flea jump many inches high.
''A dog," says Gahleo, "could probably carry two or three such
dogs upon his back; but I believe that a horse could not carry
even one of his own size."

♦ Cf. R. A. Davies and G. Fraenkel, The oxygen- consumption of flies during
flight, Jl. Exp. Biol. XVII, pp. 402-407, 1940.

t W. Thomson, On the efficiency of clothing for maintaining temperature,
Nature, xxix, p. 567, 1884.

36 ON MAGNITUDE [ch.

Such problems were admirably treated by Galileo and Borelli,
but many writers remained ignorant of their work. Linnaeus
remarked that if an elephant were as strong in proportion as a
stag-beetle, it would be able to pull up rocks and level mountains;
and Kirby and Spence have a well-known passage directed to shew
that such powers as have been conferred upon the insect have been
withheld from the higher animals, for the reason that had these
latter been endued therewith they would have "caused the early
desolation of the world*."

Such problems as that presented by the flea's jumping powersf ,
though essentially physiological in their nature, have their interest
for us here: because a steady, progressive diminution of activity
with increasing size would tend to set Hmits to the possible growth
in magnitude of an animal just as surely as those factors which
tend to break and crush the living fabric under its own weight. In
the case of a leap, we have to do rather with a sudden impulse than
with a continued strain, and this impulse should be measured in
terms of the velocity imparted. The velocity is proportional to
the impulse (x), and inversely proportional to the mass (M) moved :
V = xjM. But, according to what we still speak of as "Borelh's
law," the impulse (i.e. the work of the impulse) is proportional to
the volume of the muscle by which it is produced {, that is to say
(in similarly constructed animals) to th^ mass of the whole body;
for the impulse is proportional on the one hand to the cross-section
of the muscle, and on the other to the distance through which it

* Introduction to Entomology, ii, p. 190, 1826. Kirby and Spence, like many less
learned authors, are fond of popular illustrations of the "wonders of Nature,"
to the neglect of dynamical principles. They suggest that if a white ant were as
big as a man, its tunnels would be "magnificent cylinders of more than three
hundred feet in diameter"; and that if a certain noisy Brazilian insect were as
big as a man, its voice would be heard all the world over, "so that Stentor becomes
a mute when compared with these insects!" It is an easy consequence of
anthropomorphism, and hence a common characteristic of fairy-tales, to neglect
the dynamical and dwell on the geometrical aspect of similarity.

•f The flea is a very clever jumper; he jumps backwards, is stream-lined ac-
cordingly, and alights on his two long hind-legs. Cf. G. I. Watson, in Nature,
21 May 1938.

X That is to say, the available energy of muscle, in ft.-lbs. per lb. of muscle, is
the same for all animals: a postulate which requires considerable qualification
when we come to compare very different kinds of muscle, such as the insect's and
the mammal's.

II] OF BORELLI'S LAW 37

contracts. It follows from this that the velpcity is constant, what-
ever be the size of the animal.

Putting it still more simply, the work done in leaping is propor-
tional to the mass and to the height to which it is raised, W oc mH.
But the muscular power available for this work is proportional to
the mass of muscle, or (in similarly constructed animals) to the mass
of the animal, W oc m. It follows that H is, or tends to be, a
constant. In other words, all animals, provided always that they
are similarly fashioned, with their various levers in Hke proportion,
ought to jump not to the same relative but to the same actual
height*. The grasshopper seems to be as well planned for jumping
as the flea, and the actual heights to which they jump are much of
a muchness; but the flea's jump is about 200 times its own height,
the grasshopper's at most 20-30 times; and neither flea nor grass-
hopper is a better but rather a worse jumper than a horse or a manf .

As a matter of fact, Borelli is careful to point out that in the act
of leaping the impulse is not actually instantaneous, like the blow
of a hammer, but takes some httle time, during which the levers
are being extended by which the animal is being propelled forwards ;
and this interval of time will be longer in the case of the longer
levers of the larger animal. To some extent, then, this principle
acts as a corrective to the more general one, and tends to leave a
certain balance of advantage in regard to leaping power on the side
of the larger animal J. But on the other hand, the question of
strength of materials comes in once more, and the factors of stress
and strain and bending moment make it more and more difficult
for nature to endow the larger animal with the length of lever with
which she has provided the grasshopper or the flea. To Kirby and
Spence it seemed that "This wonderful strength of insects is
doubtless the result of something peculiar in the structure and
arrangement of their muscles, and principally their extraordinary

* Borelli, Prop. CLxxvn. Animalia minora et minus ponderosa majores saltus
efficiunt respectu sui corporis, si caetera fuerint paria.

t The high jump is nowadays a highly skilled performance. For the jumper
contrives that his centre of gravity goes under the bar, while his body, bit by bit,
goes over it.

J See also {int. al.), John Bernoulli, De Motu Musculorum, Basil., 1694;
Chabry, Mecanisme du saut, J. de VAnat. et de la Physiol, xix, 1883;* Sur la
longueur des membres des animaux sauteurs, ibid, xxi, p. 356, 1885; Le Hello,
De Taction des organes locomoteurs, etc., ibid, xxix, pp. 65-93, 1893; etc.

38 ON MAGNITUDE [ch.

power of contraction." This hypothesis, which is so easily seen on
physical grounds to be unnecessary, has been amply disproved in
a series of excellent papers by Felix Plateau*.

From the impulse of the preceding case we may pass to the momentum
created (or destroyed) under similar circumstances by a given force acting
for a given time : mv = Ft.

We know that . m oc P, and t = Ifv,

so that lH = Fllv, or v^ = F/l\

But whatsoever force be available, the animal may only exert so much of
it as is in proportion to the strength of his own limbs, that is to say to the
cross-section of bone, sinew and muscle ; and all of these cross-sections are
proportional to P, the square of the linear dimensions. The maximal force,
-f'max, which the animal dare exert is proportional, then, to l^; therefore

Fraa^JP = coustant.

And the maximal speed which the animal can safely reach, namely
F'max = ^max/^, IS also constaut, or independent {ceteris paribus) of the dimensions
of the animal.

A spurt or effort may be well within the capacity of the animal but far
beyond the margin of safety, as trainer and athlete well know. This margin
is a narrow one, whether for athlete or racehorse; both run a constant risk
of overstrain, under which they may "pull" a muscle, lacerate a tendon, or
even "break down" a bonet.

It is fortunate for their safety that animals do not jump to heights pro-
portional to their own. For conceive an animal (of mass m) to jump to
a certain altitude, such that it reaches the ground with a velocity v; then
if c be the crushing strain at any point of the sectional area (A) of the limbs,
the limiting condition is that mv = cA.

If the animal vary in magnitude without change in the height to which
it jumps (or in the velocity with which it descends), then

m P ,

coc 2 oc ^"2, orZ.

The crushing strain varies directly with the linear dimensions of the animal ;
and this, a dynamical case, is identical with the usual statical limitalfion of
magnitude.

* Recherches sur la force absolue des muscles des Inverlebres, Bull. Acad. R,
de Belgique (3), vi, vii, 1^83-84: see also ibid. (2), xx, 1865; xxii, 1866; Ann.
Mag. N.H. xvii, p. 139, 1866; xix, p. 95, 1867. Cf. M. Radau, Sur la force
musculaire des insectes, Revue des deux Mondes, lxiv, p. 770, 1866. The subject
had been well treated by Straus-Diirckheim, in his Considerations generates sur
Vanatomie comparee des animauz articuUs, 1828.

t Cf. The dynamics of sprint -running, by A. V. Hill and others, Proc. R.S. (B),
cii, pp. 29-42, 1927; or Muscular Movement in Man, by A. V. Hill, New York,
1927, ch. VI, p. 41.

II] OF LOCOMOTION 39

But if the animal, with increasing size or stature, jump to a correspondingly
increasing height, the case becomes much more serious. For the final velocity
of descent varies as the square root of the altitude reached, and therefore as
the square root of tte linear dimensions of the animal. And since, as before,

13

cccmvcc j^ Vy

c oc 75 . V ;, or c oc Z».

If a creature's jump were in proportion to its height, the crushing strains
would so increase that its dimensions would be limited thereby in a much higher
degree than was indicated by statical considerations. An animal may grow
to a size where lit is unstable dynamically, though still on the safe side
statically — a size where it moves with difficulty though it rests secure. It is
by reason of dynamical rather than of statical relations that an elephant
is of graver deportment than a. mouse.

An apparently simple problem, much less simple than it looks, lies
in the act of walking, where there will evidently be great economy of
work if the leg swing with the help of gravity, that is to say, at a
peTidulum-rate. The conical shape and jointing of the limb, the time
spent with the foot upon the ground, these and other mechanical
differences complicate the case, and make the rate hard to define or
calculate. Nevertheless, we may convince ourselves by counting our
steps, that the leg does actually tend to swing, as a pendulum does,
at a certain definite rate*. So on the same principle, but to the
slower beat of a longer pendulum, the scythe swings smoothly in
the mower's hands.

To walk quicker, we "step out"; we cause the leg-pendulum to
describe a greater arc, but it does not swing or vibrate faster until
we shorten the pendulum and begin to run. Now let two similar
individuals, A and B, walk in a similar fashion, that is to say with
a similar an^le of swing (Fig. 1). The arc through which the leg
swings, or the amplitude of each step, will then vary as the length
of leg (say as a/6), and so as the height or other linear dimension (/)
of the manf. But the time of swing varies inversely as'the square

* The assertion that the hmb tends to swing in pendulum-time was first made
by the brothers Weber {Mechanik der menschl. Gehwerkzeuge, Gottingen, 1836).
Some later writers have criticised the statement (e.g. Fischer, Die Kinematik dea
Beinschwingens etc., AhJi. math. phys. Kl. k. Sachs. Ges. xxv-xxvin, 1899-1903),
but for all that, with proper and large qualifications, it remains substantially true.

t So the stride of a Brobdingnagian was 10 yards long, or just twelve times the
2 ft. 6 in., which make the average stride or half-pace of a man.

40

ON MAGNITUDE

[CH.

root of the pendulum-length, or Va/Vb. Therefore the velocity,
which is measured by amphtude/time, or a/b x Vb/Va, will also vary

as the square root of the hnear dimen-
sions; which is Froude's law over again.
The smaller man, or smaller animal,
goes slower than the larger, but only in
the ratio of the square roots of their
linear dimensions ; whereas, if the limbs
moved alike, irrespective of the size of
the animal — if the limbs of the mouse
swung no faster than those of the horse
— then the mouse would be as slow in
its gait or slower than the tortoise.
M. DeHsle* saw a fly walk three inches
in half-a-second ; this was good steady
walking. When we walk five miles an
hour we go about 88 inches in a second,
or 88/6 =14-7 times the pace of M. DeHsle 's fly. We should walk
at just about the fly's pace if our stature were 1/(14-7)2, or 1/216
of our present height — say 72/216 inches, or one-third of an inch
high. Let us note in passing that the number of legs does not
matter, any more than the number of wheels to a coach; the
centipede runs none the faster for all his hundred legs.

But the leg comprises a complicated system of levers, by whose
various exercise we obtain very different results. For instance, by
being careful to rise upon our instep we increase the length or
amplitude of our stride, and improve our speed very materially;
and it is curious to see how Nature lengthens this metatarsal joint,
or instep-lever, in horse f and hare and greyhound, in ostrich and
in kangaroo, and in every speedy animal. Furthermore, in running
we bend and so shorten the leg, in order to accommodate it to a
quicker rate of pendulum-swing J. In short the jointed structure

* Quoted in Mr John Bishop's interesting article in Todd's Cyclopaedia, iii,
p. 443.

t The "cannon-bones" are not only relatively longer but may even be actually
longer in a little racehorse than a great carthorse.

t There is probably another factor involved here: for in bending and thus
shortening the leg, we bring its centre of gravity nearer to the pivot, that is to
say to the joint, and so the muscle tends to move it the ,more quickly. After all,

II] OF RATE OF WALKING 41

of the leg permits us to use it as the shortest possible lever while
it is swinging, and as the longest possible lever when it is exerting
its propulsive force.

The bird's case is of peculiar interest. In running, walking or
swimming, we consider the speed which an animal can attain, and
the increase of speed which increasing size permits of. But in flight
there is a certain necessary speed— a speed (relative to the air) which
the bird must attain in order to maintain itself aloft, and which nfiust
increase as its size increases. It is highly probable, as Lanchester
remarks, that Lilienthal met his untimely death (in August 1896)
not so much from any intrinsic fault in the design or construction
of his machine, but simply because his engine fell somewhat
short of the power required to give the speed necessary for its
stability.

Twenty-five years ago, when this book was written, the bird, or
the aeroplane, was thought of as a machine whose sloping wings,
held at a given angle and driven horizontally forward, deflect the
air downwards and derive support from the upward reaction. In
other words, the bird was supposed to communicate to a mass of
air a downward momentum equivalent (in unit time) to its own
weight, and to do so by direct and continuous impact. The down-
ward momentum is then proportional to the mass of air thrust
downwards, and to the rate at which it is so thrust or driven: the
mass being proportional to the wing-area and to the speed of the
bird, and the rate being again proportional to the flying speed; so
that the momentum varies as the square of the bird's linear dimen-
sions and also as the square of its speed. But in order to balance
its weight, this momentum must also be proportional to the
cube of the bird's linear dimensions; therefore the bird's necessary
speed, such as enables it to maintain level flight, must be pro-
portional to the square root of its linear dimensions, and the whole
work done must be proportional to the power 3 J of the said linear
dimensions.

The case stands, so far, as follows : m, the mass of air deflected
downwards; M, the momentum so communicated; W, the work
done — all in unit time; w, the weight, and F, the velocity of the

we know that the pendulum theory is not the whole story, but only an important
first approximation to a complex phenomenon.

42 ON MAGNITUDE [ch.

bird; I, a linear dimension, the form of the bird being supposed
constant. j^j _ ^^ ^ ^3^ but M = mV , and m = W ,

Therefore M - ^272 _ ^3^

and therefore F = VI

and Tf = MF = R

The gist of the matter is, or seems to be, that the work which
can he done varies with the available weight of muscle, that is to say,
with the mass of the bird ; but the work which has to be done varies
with mass and distance; so the larger the bird grows, the greater
the disadvantage under which all its work is done*. The dispropor-
tion does not seem very great at first sight, but it is quite enough
to tell. It is as much as to say that, every time we double the
linear dimensions of the bird, the difficulty of flight, > or the work
which must needs be done in order to fly, is increased in the ratio
of 2^ to 2^*, or 1 : V2, or say 1:14. If we take the ostrich to exceed
the sparrow in linear dimensions as 25 : 1, which seems well within
the mark, the ratio would be that between 25^* and 25^, or between
5' and 5^; in other words, flight would be five times more difficult
for the larger than for the smaller bird.

But this whole explanation is doubly inadequate. For one thing,
it takes no account of gliding flight, in which energy is drawn from
the wind, and neither muscular power nor engine power are em-
ployed; and we see that the larger birds, vulture, albatross or
solan-goose, depend on gliding more and more. Secondly, the old
simple account of the impact of the wing upon the air, and the
manner in which a downward momentum is communicated and
support obtained, is now known to be both inadequate and
erroneous. For the science of flight, or aerodynamics, has grown
out of the older science of hydrodynamics; both deal with the
special properties of a fluid, whether water or air; and in our case,
to be content to think of the air as a body of mass m, to which a
velocity v is imparted, is to neglect all its fluid properties. How the

* This is the result arrived at by Helmholtz, Ueber ein Theorem geometrisch-
ahnliche Bewegungen fliissiger Korper betrefFend, nebst Anwendung auf das
Problem Luftballons zu lenken, Monatsber. Akad. Berlin, 1873, pp. 501-514. It was
criticised and challenged (somewhat rashly) by K. Miillenhof, Die Grosse der Flug-
flachen etc., PfiUger's Archiv, xxxv, p. 407; xxxvi, p. 548, 1885.

II] OF FLIGHT 43

fish or the dolphin swims, and how the bird flies, are up to a certain
point analogous problems ; and stream-lining plays an essential part
in both. But the bird is much heavier than the air, and the fish
has much the same density as the water, so that the problem of
keeping afloat or aloft is negligible in the one, and all-important in
the other. Furthermore, the one fluid is highly compressible, and
the other (to all intents and purposes) incompressible ; and it is this
very difference which the bird, or the aeroplane, takes special
advantage of, and which helps, or even enables, it to fly.

It remains as true as ever that a bird, in order to counteract
gravity, must cause air to move downward and obtains an upward
reaction thereby. But the air displaced downward beneath the
wing accounts for a small and varying part, perhaps a third perhaps
a good deal less, of the whole force derived; and the rest is generated
above the wing, in a less simple way. For, as the air streams past
the slightly sloping wing, as smoothly as the stream-lined form
and polished surface permit, it swirls round the front or "leading"
edge*, and then streams swiftly over the upper surface of the wing;
while it passes comparatively slowly, checked by the opposing slope
of the wing, across the lower side. And this is as much as to say
that it tends to be compressed below and rarefied above; in other
words, that a partial vacuum is formed above the wing and follows
it wherever it goes, so long as the stream-lining of the wing and its
angle of incidence are suitable, and so long as the bird travels fast
enough through the air.

The bird's weight is exerting a downward force upon the air, in
one way just as in the other; and we can imagine a barometer
delicate enough to shew and measure it as the bird flies overhead.
But to calculate that force we should have to consider a multitude
of component elements; we should have to deal with the stream-
lined tubes of flow above and below, and the eddies round the fore-
edge of the wing and elsewhere; and the calculation which was too
simple before now becomes insuperably difficult. But the principle
of necessary speed remains as true as ever. The bigger the bird

* The arched form, or "dipping front edge" of the wing, and its use in causing
a vacuum above, were first recognised by Mr H. F. Phillips, who put the idea into
a patent in 1884. The facts were discovered independently, and soon afterwards,
both by Lilienthal and Lanchester.

44 ON MAGNITUDE [ch.

becomes, the more swiftly must the air stream over the wing
to give rise to the rarefaction or negative pressure which is more
and more required; and the harder must it be to fly, so long as
work has to be done by the muscles of the bird. The general
principle is the same as before, though the quantitative relation
does not work out as easily as it did. As a matter of fact, there
is probably little difference in the end; and in aeronautics, the
"total resultant force" which the bird employs for its support is
said, e^npirically, to vary as the square of the air-speed: which is
then a result analogous to Froude's law, and is just what we arrived
at before in the simpler and less accurate setting of the case.

But a comparison between the larger and the smaller bird, like
all other comparisons, applies only so long as the other factors in
the case remain the same ; and these vary so much in the complicated
action of flight that it is hard indeed to compare one bird with
another. For not only is the bird continually changing the incidence
of its wing, but it alters the lie of every single important feather;
and all the ways and means of flight vary so enormously, in big
wings and small, and Nature exhibits so many refinements and
" improvements" in the mechanism required, that a comparison based
on size alone becomes imaginary, and is little worth the making.

The above considerations are of great practical importance in
aeronautics, for they shew how a provision of increasing speed must ac-
company every enlargement of our aeroplanes. Speaking generally,
the necessary or minimal speed of an aeroplane varies as the square
root of its Unear dimensions; if (ceteris paribus) we make it four
times as long, it must, in order to remain aloft, fly twice as fast as
before*. If a given machine weighing, say, 500 lb. be stable at
40 miles an hour, then a geometrically similar one which weighs,
say, a couple of tons has its speed determined as follows :

W:w::L^:l^::S:l.

Therefore L:l::2:l.

But V^:v^::L:l

Therefore V:v::V2:l = 1-414 : 1.

* G. H. Bryan, Stability in Aviation, 1911; F. W. Lanchester, Aerodynamics,
1909; cf. (int. al.) George Greenhill, The Dynamics of Mechanical Flight, 1912;
F. W. Headley, The Flight of Birds, and recent works.

II] OF NECESSARY SPEED 45

That is to say, the larger machine must be capable of a speed of
40 X 1-414, or about 56|, miles per hour.

An arrow is a somewhat rudimentary flying-machine; but it is
capable, to a certain extent and at a high velocity, of acquiring
"stability," and hence of actual flight after the fashion of an aero-
plane; the duration and consequent range of its trajectory are
vastly superior to those of a bullet of the same initial velocity.
Coming back to our birds, and again comparing the ostrich with
the sparrow, we find we know little or nothing about the actual
speed of the latter ; but the minimal speed of the swift is estimated
at 100 ft. per second, or even more — say 70 miles an hour. We
shall be on the safe side, and perhaps not far wrong, to take 20 miles
an hour as the sparrow's minimal speed; and it would then follow
that the ostrich, of 25 times the sparrow's linear dimensions, would
have to fly (if it flew at all) with a minimum velocity of 5 x 20,
or 100 miles an hour*.

The same principle of necessary speed, or the inevitable relation
between the dimensions of a flying object and the minimum velocity
at which its flight is stable, accounts for a considerable number of

* Birds have an ordinary and a forced speed. Meinertzhagen puts the ordinary
flight of the swift at 68 m.p.h., which talhes with the old estimate of Athanasius
Kircher (Physiologia, ed. 1680, p. 65) of 100 ft. per second for the swallow. Abel
Chapman {Retrospect, 1928, ch. xiv) puts the gliding or swooping flight of the swift
at over 150 m.p.h., and that of the griffon vulture at 180 m.p.h.; but these skilled
fliers doubtless far exceed the necessary minimal speeds which we are speaking of.
An airman flying at 70 m.p.h. has seen a golden eagle fly past him easily; but
even this speed is exceptional. Several observers agree in giving 50 m.p.h. for
grouse and woodcock, and 30 m.p.h. for starling, chaffinch, quail and crow. A
migrating flock of lapwing travelled at 41 m.p.h., ten or twelve miles more than
the usual speed of the single bird. Lanchester, on theoretical considerations,
estimates the speed of the herring gull at 26 m.p.h., and of the albatross at about
34 miles. A tern, a very skilful flier, was seen to fly as slowly as 15 m.p.h.
A hornet or a large dragonfly may reach 14 or 18 m.p.h.; but for most insects
2-4 metres per sec, say 4-9 m.p.h., is a common speed (cf, A. Magnan, Vol.
des Insectes, 1834, p. 72). The larger diptera are very swift, but their speed is much
exaggerated. A deerfly (Cephenomyia) has been said to fly at 400 yards per second,
or say 800 m.p.h., an impossible velocity (Irving Langmuir, Science, March 11, 1938).
It would mean a pressure on the fly's head of half an atmosphere, probably enough
to crush the fly; to maintain it would take half a horsepower; and this would need
a food- consumption of 1^ times the fly's weight per second\ 25 m.p.h. is a more
reasonable estimate. The naturalist should not forget, though it does not touch
our present argument, that the aeroplane is built to the pattern of a beetle rather
than of a bird; for the elytra 'are not wings but planes. Cf. int. ah, P. Amans,
Geometrie. . .des ailes rigides, C.R. Assoc. Frang. pour Vavancem. des Sc. 1901.

46 ON MAGNITUDE [ch,

observed phenomena. It tells us why the larger birds have a
marked difficulty in rising from the ground, that is to say, in
acquiring to begin with the horizontal velocity necessary for their
support; and why accordingly, as Mouillard* and others have
observed, the heavier birds, even those weighing no more than a
pound or two, can be effectually caged in small enclosures open
to the sky. It explains why, as Mr Abel Chapman says, "all
ponderous birds, wild swans and geese, great bustard and caper-
cailzie, even blackcock, fly faster than they appear to do," while
"light-built types with a big wing-areaf, such as herons and harriers,
possess no turn of speed at all." For the fact is that the heavy
birds must fly quickly, or not at all. It tells us why very small
birds, especially those as small as humming-birds, and a fortiori the
still smaller insects, are capable of "stationary flight," a very slight
and scarcely perceptible velocity relatively to the air being sufficient
for their support and stabihty. And again, since it is in all
these cases velocity relatively to the air which we are speaking
of, we comprehend the reason why one may always tell which
way the wind blows by watching the direction in which a bird starts
to fly.

The wing of a bird or insect, like the tail of a fish or the blade
of an oar, gives rise at each impulsion to a swirl or vortex, which
tends (so to speak) to chng to it and travel along with it;' and the
resistance which wing or oar encounter comes much more from
these vortices than from the viscosity of the fluid. J We learn as a
corollary to this, that vortices form only at the edge of oar or wing —
it is only the length and not the breadth of these which matters.
A long narrow oar outpaces a broad one, and the efficiency of the
long, narrow wing of albatross, swift or hawkmoth is so far accounted
for. From the length of the wing we can calculate approximately
its rate of swing, and more conjecturally the dimensions of each
vortex, and finally the resistance or Ufting power of the stroke;
and the result shews once again the advantages of the small-scale

* Mouillard, L' empire de Vair; essai d'ornithologie appliqu4e a Vaviationf 1881;
transl. in Annual Report of the Smithsonian Institution^ 1892.

t On wing-area in relation to weight of bird see Lendenfeld in Naturw. Wochenschr.
Nov. 1904, transl. in Smithsonian Inst. Rep. 1904; also E. H. Hankin, Animal
Flight, 1913; etc.

X Cf. V. Bjerknes, Hydrodynamique physique, n, p. 293, 1934.

II] OF MODES OF FLIGHT 47

mechanism, and the disadvantage under which the larger machine
or larger creature hes.

Length

Speed of

Radius

Force of

Specific

Weight

of wing Beats

wing-tip

of

wing-beat

j orce,

gm.

m. per sec
(From V.

m./s.
Bjerknes)

vortex*

gra.

FjW

Stork

3500

0-91 2

5-7

1-5

1480

2:5

Gull

1000

0-60 3

5-7

1-0

640

2:3

Pigeon

350

0-30 6

5-7

0-5

160

1:2

Sparrow

30

Oil 13

4-5

0-18

13

2:5

Bee

0-07

0-01 200

6-3

002

0-2

3i:l

Fly

001

0-007 190

4-2

001

004

4:1

* Conjectural.

A bird may exert a force at each stroke of its wing equal to
one-half, let us say for safety one-quarter, of its own weight, more
or less ; but a bee or a fly does twice or thrice the equivalent of its
own weight, at a low estimate. If stork, gull or pigeon can thus
carry only one-fifth, one-third, one- quarter of their weight by the
beating of their wings, it follows that all the rest must be borne by
sailing-flight between the wing-beats. But an insect's wings lift it
easily and with something to spare; hence saihng-flight, and with
it the whole principle of necessary speed, does not concern the lesser
insects, nor the smallest birds, at all; for a humming-bird can
"stand still" in the air, like a hover-fly, and dart backwards as
well as forwards, if it please.

There is a little group of Fairy-flies (Mymaridae), far below the
size of any small famiUar insects; their eggs are laid and larvae
reared within the tiny eggs of larger insects ; their bodies may be no
more than J mm. long, and their outspread wings 2 mm. from tip
to tip (Fig. 2). It is a pecuharity of some of these that their Httle
wings are made of a few hairs or bristles, instead of the continuous
membrane of a wing. How these act on the minute quantity of air
involved we can only conjecture. It would seem that that small
quantity reacts as a viscous fluid to the beat of the wing ; but there
are doubtless other unobserved anomahes in the mechanism and
the mode of flight of these pigmy creaturesf .

The ostrich has apparently reached a magnitude, and the moa
certainly did so, at which flight by muscular action, according to

t It is obvious that in a still smaller order of magnitude tfie Brownian movement
would suffice to make^ flight impossible.

48

ON MAGNITUDE

[CH.

the normal anatomy of a bird, becomes physiologically impossible.
The same reasoning applies to the case of man. It would be very
difficult, and probably absolutely impossible, for a bird to flap its
way through the air were it of the bigness of a man; but Borelli,
in discussing the matter, laid even greater stress on the fact that
a man's pectoral muscles are so much less in proportion than those
of a bird, that however we might fit ourselves out with wings, we
could never expect to flap them by any power of our own weak
muscles. Borelli had learned this lesson thoroughly, and in one of
his chapters he deals with the proposition : Est impossibile ut homines
jyropriis viribus artificiose volare possinV^, But gliding flight, where

a ' b

2. Fairy-flies (Mymaridae) : after F. Enock. x 20.

wind-force and gravitational energy take the place of muscular
power, is another story, and its limitations are of another kind.
Nature has many modes and mechanisms of flight, in birds of one
kind and another, in bats and beetles, butterflies, dragonflies and
what not ; and gliding seems to be the common way of birds, and
the flapping flight {remigio alarum) of sparrow and of crow to be
the exception rather than the rule. But it were truer to say that
gliding and soaring, by which energy is captured from the wind, are
modes of flight little needed by the small birds, but more and more
essential to the large. BorelH had proved so convincingly that
we could never hope to fly propriis viribus, that all through the
eighteenth century men tried no more to fly at all. It was in trying
to glide that the pioneers of aviation, Cayley, Wenham and Mouillard,

* Giovanni Alfonso Borelli, De Motu Animalium, i, Prop, cciv, p. 243, edit.
1685. The part on The Flight of Birds is issued by the Royal Aeronautical Society
as No. 6 of its Aeronautical Classics.

II] OF GLIDING FLIGHT 49

Langley, Lilienthal and the Wrights — all careful students of birds —
renewed the attempt*; and only after the Wrights had learned to
ghde did they seek to add power to their glider. Flight, as the
Wrights declared, is a matter of practice and of skill, and skill in
gliding has now reached a point which more than justifies all
Leonardo da Vinci's attempts to fly. Birds shew infinite skill and
instinctive knowledge in the use they make of the horizontal accelera-
tion of the wind, and the advantage they take of ascending currents
in the air. Over the hot sands of the Sahara, where every here
and there hot air is going up and cooler coming down, birds keep
as best they can to the one, or ghde quickly through the other;
so we may watch a big dragonfly planing slowly down a few feet
above the heated soil, and only every five minutes or so regaining
height with a vigorous stroke of his wings. The albatross uses the
upward current on the lee-side of a great ocean- wave ; so, on a lesser
scale, does the flying- fish; and the seagull flies in curves, taking
every advantage of the varying wind- velocities at different levels
over the sea. An Indian vulture flaps his way up for a few laborious
yards, then catching an upward current soars in easy spirals to
2000 feet ; here he may stay, effortless, all day long, and come down
at sunset. Nor is the modern sail-plane much less efiicient than a
soaring bird ; for a skilful pilot in the tropics should be able to roam
all day long at willf .

A bird's sensitiveness to air-pressure is indicated in other ways
besides. Heavy birds, hke duck and partridge, fly low and ap-
parently take advantage of air-pressure reflected from the ground.
Water-hen and dipper follow the windings of the stream as they fly
up or down; a bee-hne would give them a shorter course, but not
so smooth a journey. Some small birds — wagtails, woodpeckers and
a few others — fly, so to speak, by leaps and bounds ; they fly briskly

* Sir George Cayley (1774-1857), father of British aeronautics, was the first to
perceive the capabilities of rigid planes, and to experiment on gliding flight. He
anticipated all the essential principles of the modern aeroplane, and his first paper
"On Aerial Navigation" appeared in Nicholson's Journal for November 1809.
F. H. Wenham (1824-1908) studied the flight of birds and estimated the necessary
proportion of surface to weight and speed; he held that "the whole secret of
success in flight depends upon a proper concave form of the supporting surface."
See his paper "On Aerial Locomotion" in the Report of the Aeronautical Society
1866.

t Sir Gilbert Walker, in Nature, Oct. 2, 1937.

50 ON MAGNITUDE [ch.

for a few moments, then close their wings and shoot along*. The
flying-fishes do much the same, save that they keep their wings
outspread. The best of them "taxi" along with only their tails in
the water, the tail vibrating with great rapidity, and the speed
attained lasts the fish on its long glide through the airf.

Flying may have begun, as in Man's case it did, with short spells
of gUding flight, helped by gravity, and far short of sustained or
continuous locomotion. The short wings and long tail of Archae-
opteryx would be efficient as a slow-speed ghder; and we may still
see a Touraco glide down from his perch looking not much unlike
Archaeopteryx in the proportions of his wings and tail. The small
bodies, scanty muscles and narrow but vastly elongated wings of
a Pterodactyl go far beyond the hmits of mechanical efficiency for
ordinary flapping flight; but for ghding they approach perfection J.
Sooner or later Nature does everything which is physically possible ;
and to glide with skill and safety through the air is a possibihty
which she did not overlook.

Apart from all differences in the action of the limbs — apart from
differences in mechanical construction or in the manner in which
the mechanism is used — we have now arrived at a curiously simple
and uniform result. For in all the three forms of locomotion which
we have attempted to study, alike in swimming and in walking, and
even in the more, complex problem of flight, the general result,
obtained under very different conditions and arrived at by different
modes of reasoning, shews in every case that speed tends to vary as
the square root of the linear dimensions of the animal.

While the rate of progress tends to increase slowly with increasing
size (according to Froude's law), and the rhythm or pendulum-rate
of the limbs to increase rapidly with decreasing size (according to
Galileo's law), some such increase of velocity with decreasing

* Why large birds cannot do the same is discussed by Lanchester, op. cit.
Appendix iv.,

t Cf. Carl L. Hubbs, On the flight of. . .the Cypselurinae, and remarks on the
evolution of the flight of fishes. Papers of the Michigan Acad, of Sci. xvii, pp. 575-
611, 1933. See also E. H. Hankin, P.Z.S. 1920, pp. 467-474; and C. M. Breeder,
On the structural specialisation of flying fishes from the standpoint of aero-
dynamics, Copeia, 1930, pp. 114-121.

X The old conjecture that their flight was helped or rendered possible by a denser
atmosphere than ours is thus no longer called for.

II] OF THE FORCE OF GRAVITY 51

magnitude is true of all the rhythmic actions of the body, though
for reasons not always easy to explain. The elephant's heart beats
slower than ours*, the dog's quicker; the rabbit's goes pit-a-pat;
the mouse's and the sparrow's are too quick to count. But the very
"rate of Hving" (measured by the consumed and COg produced)
slows down as size increases; and a rat lives so much faster than
a man that the years of its life are three, instead of threescore and
ten.

From all the foregoing discussion we learn that, as Crookes once
upon a time remarked I, the forms as well as the actions of our
bodies are entirely conditioned (save for certain exceptions in the
case of aquatic animals) by the strength of gravity upon this globe;
or, as Sir Charles Bell had put it some sixty years before, the very
animals which move upon the surface of the earth are proportioned
to its magnitude. Were the force of gravity to be doubled our
bipedal form would be a failure, and the majority of terrestrial
animals would resemble short-legged saurians, or else serpents.
Birds and insects would suffer Ukewise, though with some com-
pensation in the increased density of the air. On the other hand,
if gravity were halved, we should get a lighter, slenderer, more active
type, needing less energy, less heat, less heart, less lungs, less blood.
Gravity not only controls the actions but also influences the forms
of all save the least of organisms. The tree under its burden of
leaves or fruit has changed its every curve and outline since its
boughs were bare, and a mantle of snow will alter its configuration
again. Sagging wrinkles, hanging breasts and many another sign
of age are part of gravitation's slow relentless handiwork.

There are other physical factors besides gravity which help to
limit the size to which an animal may grow and to define the con-
ditions under which it may Hve. The small insects skating on a
pool have their movements controlled and their freedom hmited by
the surface-tension between water and air, and the measure of that
tension determines the magnitude which they may attain. A man
coming wet from his bath carries a few ounces of water, and is
perhaps 1 per cent, heavier than before; but a wet fly weighs twice
as much as a dry one, and becomes a helpless thing. * A small

* Say 28 to 30 beats to the minute.

t Proc. Psychical Soc. xn, p. 338-355, 1^97.

52 ON MAGNITUDE [ch.

insect finds itself imprisoned in a drop of water, and a fly with
two feet in one drop finds it hard to extricate them.

The mechanical construction of insect or crustacean is highly
efficient up to a certain size, but even crab and lobster never exceed
certain moderate dimensions, perfect within these narrow bounds as
their construction seems to be. Their body lies within a hollow
shell, the stresses within which increase much faster than the mere
scale of size ; every hollow structure, every dome or cyhnder, grows
weaker as it grows larger, and a tin canister is easy to make but a
great boiler is a complicated affair. The boiler has to be strengthened
by "stiffening rings" or ridges, and so has the lobster's shell; but
there is a limit even to this method of counteracting the weakening
effect of size. An ordinary girder-bridge may be made»efficient up
to a span of 200 feet or so ; but it is physically incapable of spanning
the Firth of Forth. The great Japanese spider-crab, Macrocheira,
has a span of some 12 feet across; but Nature meets the difficulty
and solves the problem by keeping the body small, and building up
the long and slender legs out of short lengths of narrow tubes.
A hollow shell is admirable for small animals, but Nature does not
and cannot make use of it for the large.

In the case of insects, other causes help to keep them of small
dimensions. In their peculiar respiratory system blood does not
carry oxygen to the tissues, but innumerable fine tubules or tracheae
lead air into the interstices of the body. If we imagine them growing
even to the size of crab or lobster, a vast complication of tracheal
tubules would be necessary, within which friction would increase
and diffusion be retarded, and which would soon be an inefficient
and inappropriate mechanism.

The vibration of vocal chords and auditory drums has this in
common with the pendulum-hke motion of a hmb that its rate
also tends to vary inversely as the square root of the linear dimen-
sions. We know by common experience of fiddle, drum or organ,
that pitch rises, or the frequency of vibration increases, as the
dimensions of pipe or membrane or string diminish; and in like
manner we expect to hear a bass note from the great beasts and a
piping treble from the small. The rate of vibration {N) of a stretched
string depends on its tension and its density; these beins^ equal, it
varies inversely as its own length and as its diameter, i^or similar

II] OF EYES AND EARS 53

strings, N oc 1//^, and for a circular membrane, of radius r and
thickness e, N oc ll(r^ Ve).

But the deHcate drums or tympana of various animals seem to
vary much less in thickness than in diameter, and we may be content
to write, once more, N oc l/r^.

Suppose one animal to be fifty times less than another, vocal
chords and all: the one's voice will be pitched 2500 times as many
beats, or some ten or eleven octaves, above the other's; and the
same comparison, or the same contrast, will apply to the tympanic
membranes by which the vibrations are received. But our own
perception of musical notes only reaches to 4000 vibrations per
second, or thereby; a squeaking mouse or bat is heard by few, and
to vibrations of 10,000 per second we are all of us stone-deaf.
Structure apart, mere size is enough to give the lesser birds and
beasts a music quite different to our own: the humming-bird, for
aught we know, may be singing all day long. A minute insect may
utter and receive vibrations of prodigious rapidity; even its little
wings may beat hundreds of times a second*. Far more things
happen to it in a second than to us ; a thousandth part of a second
is no longer neghgible, and time itself seems to run a different course
to ours.

The eye and its retinal elements have ranges of magnitude and
Hmitations of magnitude of their own. A big dog's eye is hardly
bigger than a little dog's; a squirrel's is much larger, propor-
tionately, than an elephant's; and a robin's is but little less than
a pigeon's or a crow's. For the rods and cones do not vary with
the size of the animal, but have their dimensions optically limited
by the interference-patterns of the waves of light, which set bounds
to the production of clear retinal images. True, the larger animal
may want a larger field of view ; but this makeg little difference, for
but a small area of the retina is ever needed or used. The eye, in
short, can never be very small and need never be very big; it has
its own conditions and limitations apart from the size of the animal.
But the insect's eye tells another story. If a fly had an eye like
ours, the pupil would be so small that diffraction would render a
clear image impossible. The only alternative is to unite a number

* The wing-beats are said to be as follows: dragonfly 28 per sec, bee 190,
housefly 330; cf. Erhard, Verh. d. d. zool. Gesellsch. 1913, p. 206.

54 ON MAGNITUDE [ch.

of small and optically isolated simple eyes into a compound eye,
and in the insect Nature adopts this alternative possibiHty*.

Our range of vision is limited to a bare octave of "luminous"
waves, which is a considerable part of the whole range of Hght-heat
rays emitted by the sun; the sun's rays extend into the ultra-violet
for another half-octave or more, but the rays to which our eyes are
sensitive are just those which pass with the least absorption through
a watery medium. Some ancient vertebrate may have learned to
see in an ocean which let a certain part of the sun's whole radiation
through, which part is our part still ; or perhaps the watery media
of the eye itself account sufficiently for the selective filtration. In
either case, the dimensions of the retinal elements are so closely
related to the wave-lengths of light (or to their interference patterns)
that we have good reason to look upon the retina as perfect of its
kind, within the hmits which the properties of hght itself impose;
and this perfection is further illustrated by the fact that a few
light-quanta, perhaps a single one, suffice to produce a sensation "j".
The hard eyes of insects are sensitive over a wider range. The bee
has two visual optima, one coincident with our own, the other and
principal one high up in the ultra-violet J. And with the latt^er the
bee is able to see that ultra-violet which is so well reflected by many
flowers that flower-photographs have been taken through a filter
which passes these but transmits no other rays§.

When we talk of hght, and of magnitudes whose order is that of
a wave-length of hght, the subtle phenomenon of colour is near at
hand. The hues of hving things are due to sundry causes; where
they come from chemical pigmentation they are outside our theme,
but oftentimes there is no pigment at all, save perhaps as a screen
or background, and the tints are those proper to a scale of wave-
lengths or range of magnitude. In birds these "optical colours"
are of two chief kinds. One kind include certain vivid blues, the

* Cf. C. J. van der Horst, The optics of the insect eye, Acta Zoolog. 1933,
p. 108.

t Cf. Niels Bohr, in Nature, April 1, 1933, p. 457. Also J. Joly, Proc. R.S. (B),
xcn, p. 222, 1921.

f L. M. Bertholf, Reactions of the honey-bee to light, Journ. of Agric. Res.
XLm, p. 379; xliv, p. 763, 1931.

§ A. Kuhn, Ueber den Farbensinn der Bienen, Ztschr. d. vergl. Physiol, v,
pp. 762-800, 1927; cf. F. K. Richtmeyer, Reflection of ultra-violet by flowers,
J num. Optical Soc. Amer. vii, pp. 151-168, 1923; etc.

II] OF LIGHT AND COLOUR 55

blue of a blue jay, an Indian roller or a macaw; to the other belong
the iridescent hues of mother-of-pearl, of the humming-bird, the
peacock and the dove: for the dove's grey breast shews many
colours yet contains but one — colores inesse plures nee esse plus uno,
as Cicero said. The jay's blue feather shews a layer of enamel-like
cells beneath a thin horny cuticle, and the cell-walls are spongy
with innumerable tiny air-filled pores. These are about 0-3 /it in
diameter, in some birds even a little less, and so are not far from
the hmits of microscopic vision. A deeper layer carries dark-brown
pigment, but there is no blue pigment at all; if the feather be dipped
in a fluid of refractive index equal to its own, the blue utterly
disappears, to reappear when the feather dries. This blue is like
the colour of the sky; it is ''Tyndall's blue," such as is displayed
by turbid media, cloudy with dust-motes or tiny bubbles of a size
comparable to the wave-lengths of the blue end of the spectrum.
The longer waves of red or yellow pass through, the shorter violet
rays are reflected or scattered; the intensity of the blue depends
on the size and concentration of the particles, while the dark pigment-
screen enhances the effect.

Rainbow hues are more subtle and more complicated ; but in the
peacock and the humming-bird we know for certain* that the
colours are those of Newton's rings, and are produced by thin plates
or films covering the barbules of the feather. The colours are such
as are shewn by films about J [x thick, more or less ; they change
towards the blue end of the spectrum as thei hght falls more and
more obliquely; or towards the red end if you soak the feather
and cause the thin plates to swell. The barbules of the peacock's
feather are broad and flat, smooth and shiny, and their cuticular
layer sphts into three very thin transparent films, hardly more than
1 jjL thick, all three together. The gorgeous tints of the humming-
birds have had their places in Newton's scale defined, and the
changes which they exhibit at varying incidence have been predicted

* Rayleigh, Phil. Mag. (6), xxxvii, p. 98, 1919. For a review of the whole
subject, and a discussion of its many difficulties, see H. Onslow, On a periodic
structure in many insect scales, etc., Phil. Trans. (B), ccxi, pp. 1-74, 1921;
also C. W. Mason, Journ. Physic. Chemistry, xxvii, xxx, xxxi, 1923-25-27;
F. Suffert, Zeitschr. f. Morph. u. Oekol. d. Tiere, i, pp. 171-306, 1924 (scales of
butterflies); also B. Reusch and Th. Elsasser in Journ. f. Ornithologie, lxxiti,
1925; etc.

56 ON MAGNITUDE [ch.

and explained. The thickness of each film lies on the very limit of
microscopic vision, and the least change or irregularity in this
minute dimension would throw the whole display of colour out of
gear. No phenomenon of organic magnitude is more striking than
this constancy of size; none more remarkable than that these fine
lamellae should have their tenuity so sharply defined, so uniform
in feather after feather, so identical in all the individuals of a species,
so constant from one generation to another.

A simpler phenomenon, and one which is visible throughout the
whole field of morphology, is the tendency (referable doubtless in
each case to some definite physical cause) for mere bodily surface
to keep pace with volume, through some alteration of its form. The
development of villi on the lining of the intestine (which increase
its surface much as we enlarge the effective surface of a bath-towel),
the various valvular folds of the intestinal lining, including the
remarkable "spiral valve" of the shark's gut, the lobulation of the
kidney in large animals*, the vast increase of respiratory surface in
the air-sacs and alveoli of the lung, the development of gills in the
larger Crustacea and worms though the general surface of the body
suffices for respiration in the smaller species — all these and many
more are cases in which a more or less constant ratio tends to be
maintained between mass and surface, which ratio would have been
more and more departed from with increasing size, had it not been
for such alteration of surface-form f. A leafy wood, a grassy sward,
a piece of sponge, a reef of coral, are all instances of a hke pheno-
menon. In fact, a deal of evolution is involved in keeping due
balance between surface and mass as growth goes on.

In the case of very small animals, and of individual cells, the
principle becomes especially important, in consequence of the
molecular forces whose resultant action is limited to the superficial
layer. In the cases just mentioned, action is facilitated by increase
of surface : diffusion, for instance, of nutrient liquids or respiratory
gases is rendered more rapid by the greater area of surface; but

* Cf. R. Anthony, C.R. clxix, p. 1174, 1919, etc. Cf. also A. Putter, Studien
uber physiologische Ahnlichkeit, Pfluger's Archiv, clxviii, pp. 209-246, 1917.

t For various calculations of the increase of surface due to histological and
anatomical subdivision, see E. Babak, Ueber die Oberflachenentwickelung bei
Organismen, Biol. Centralbl. xxx, pp. 225-239, 257-267, 1910.

II] OF SURFACE AND VOLUME 57

there are other cases in which the ratio of surface to mass may
change the whole condition of the system. Iron rusts when exposed
to moist air, but it rusts ever so much faster, and is soon eaten away,
if the iron be first reduced to a heap of small filings ; this is a mere
difference of degree. But the spherical surface of the rain-drop
and the spherical surface of the ocean (though both happen to be
ahke in mathematical form) are two totally different phenomena,
the one due to surface-energy, and the other to that form of mass-
energy which we ascribe to gravity. The contrast is still more
clearly seen in the case of waves: for the little ripple, whose form
and manner of propagation are governed by surface-tension, is
found to travel with a velocity which is inversely as the square
root of its length; while the ordinary big waves, controlled by
gravitation, have a velocity directly proportional to the square root
of their wave-length. In hke manner we shall find that the form
of all very small organisms is independent of gravity, and largely
if not mainly due to the force of surface-tension: either as the
direct result of the continued action of surface-tension on the
semi-fluid body, or else as the result of its action at a prior stage
of development, in bringing about a form which subsequent chemical
changes have rendered rigid and lasting. In either case, we shall
find a great tendency in small organisms to assume either the
spherical form or other simple forms related to ordinary inanimate
surface-tension phenomena, which forms do not recur in the
external morphology of large animals.

Now this is a very important matter, and is a notable illustration
of that principle of simihtude which we have already discussed in
regard to several of its manifestations. We are coming to a con-
clusion which will affect the whole course of our argument throughout
this book, namely that there is an essential difference in kind
between the phenomena of form in the larger and the smaller
organisms. I have called this book a study of Growth and Fonn,
because in the most familiar illustrations of organic form, as in our
own bodies for example, these two factors are inseparably asso-
ciated, and because we are here justified in thinking of form as the
direct resultant and consequence of growth: of growth, whose
varying rate in one direction or another has produced, by its gradual
and unequal increments, the successive stages of development and

58 ON MAGNITUDE [ch.

the final configuration of the whole material structure. But it is
by no means true that form and growth are in this direct and simple
fashion correlative or complementary in the case of minute portions
of living matter. For in the smaller organisms, and in the indi-
vidual cells of the larger, we have reached an order of magnitude
in which the intermolecular forces strive under favourable conditions
with, and at length altogether outweigh, the force of gravity, and
also those other forces leading to movements of convection which
are the prevailing factors in the larger material aggregate.

However, we shall require to deal more fully with this matter in
our discussion of the rate of growth, and we may leave it mean-
while, in order to deal with other matters more or less directly
concerned with the magnitude of the cell.

The hving cell is a very complex field of energy, and of energy
of many kinds, of which surface-energy is not the least. Now the
whole surface-energy of the cell is by no means restricted to its
outer surface; for the cell is a very heterogeneous structure, and all
its protoplasmic alveoli and other visible (as well as invisible) hetero-
geneities make up a great system of internal surfaces, at every part
of which one "phase" comes in contact with another "phase," and
surface-energy is manifested accordingly. But still, the external
surface is a definite portion of the system, with a definite "phase"
of its own, and however little we may know of the distribution of
the total energy of the system, it is at least plain that the conditions
which favour equihbrium will be greatly altered by the changed
ratio of external surface to mass which a mere change of magnitude
produces in the cell. In short, the phenomenon of division of the
growing cell, however it be brought about, will be precisely what
is wanted to keep fairly constant the ratio between surface and
mass, and to retain or restore the balance between surface-energy
and the other forces of the system*. But when a germ-cell divides
or "segments" into two, it does not increase in mass; at least if
there be some shght alleged tendency for the egg to increase in

* Certain cells of the cucumber were found to divide when they had grown to
a volume half as large again as that of the "resting cells." Thus the volumes
of resting, dividing and daughter cells were as 1:1-5: 0-75; and their surfaces,
being as the power 2/3 of these figures, were, roughly, as 1:1-3: 0-8. The ratio
of SjV was then as 1 : 0-9 : 1-1, or much nearer equality. Cf. F. T. Lewis, Anat.
Record, xlvii, pp. 59-99, 1930.

II] OF THE SIZE OF DROPS 59

mass or volume during segmentation it is very slight indeed,
generally imperceptible, and wholly denied by some*. The growth
or development of the egg from a one-celled stage to stages of two
or many cells is thus a somewhat pecuhar kind of growth; it is
growth limited to change of form and increase of surface, unaccom-
panied by growth in volume or in mass. In the case of a soap-bubble,
by the way, if it divide into two bubbles the volume is actually
diminished, while the surface-area is greatly increased! ; the diminution
being due to a cause which we shall have to study later, namely to
the increased pressure due to the greater curvature of the smaller
bubbles.

An immediate and remarkable result of the principles just
described is a tendency on the part of all cells, according to their
kind, to vary but little about a certain mean size, and to have in
fact certain absolute limitations of magnitude. The diameter of a
large parenchymatous cell is perhaps tenfold that of a httle one;
but the tallest phanerogams are ten thousand times the height of
the least. In short, Nature has her materials of predeterminate
dimensions, and keeps to the same bricks whether she build a great
house or a small. Even ordinary drops tend towards a certain
fixed size, which size is a function of the surface-tension, and may
be used (as Quincke used it) as a measure thereof. In a shower of
rain the principle is curiously illustrated, as Wilding Roller and
V. Bjerknes tell us. The drops are of graded sizes, each twice as big
as another, beginning with the minute and uniform droplets of an
impalpable mist. They rotate as they fall, and if two rotate in
contrary directions they draw together and presently coalesce; but
this only happens when two drops are faUing side by side, and since
the rate of fall depends on the size it always is a pair of coequal
drops which so meet, approach and join together. A supreme
instance of constancy or quasi-constancy of size, remote from but
yet analogous to the size-hmitation of a rain-drop or a cell, is the
fact that the stars of heaven (however else one diifereth from
another), and even the nebulae themselves, are all wellnigh co-equal
in mass. Gravity draws matter together, condensing it into a world

* Though the entire egg is not increasing in mass, that is not to say that its living
protoplasm is not increasing all the while at the expense of the reserve material,
t Cf. P. G. Tait, Proc. E.S.E. v, 1866 and vi, 1868.

60 ON MAGNITUDE [ch.

or into a star; but ethereal pressure is an opponent force leading
to disruption, negligible on the small scale but potent on the large.
High up in the scale of magnitude, from about 10^^ to 10^^ grams
of matter, these two great cosmic forces balance one another; and
all the magnitudes of all the stars he within or hard by these narrow
limits.

In the hving cell, Sachs pointed out (in 1895) that there is a
tendency for each nucleus to gather around itself a certain definite
amount of protoplasm*. Drieschf, a httle later, found it possible,
by artificial subdivision of the egg, to rear dwarf sea-urchin larvae,
one-half, one-quarter or even one-eighth of their usual size; which
dwarf larvae were composed of only a half, a quarter or an eighth
of the normal number of cells. These observations have been often
repeated and amply confirmed : and Loeb found the sea-urchin eggs
capable of reduction to a certain size, but no further.

In the development of Crepidula (an American "shpper-Hmpet,"
now much at home on our oyster-beds), ConklinJ has succeeded in
rearing dwarf and giant individuals, of which the latter may be
five-and-twenty times as big as the former. But the individual
cells, of skin, gut, liver, muscle and other tissues, are just the same
size in one as in the other, in dwarf and in giant §. In like manner

* Physiologische Notizen (9), p. 425, 1895. Cf. Amelung, Flora, 1893; Stras-
biirger, Ueber die Wirkungssphare der Kerne und die Zellgrosse, Histol. Beitr. (5),
pp. 95-129, 1893; R. Hertwig, Ueber Korrelation von Zell- und Kerngrosse
(Kernplasraarelation), Biol. Centralbl. xvm, pp. 49-62, 108-119, 1903; G. Levi
and T. Terni, Le variazioni dell' indice plasmatico-nucleare durante 1' intercinesi,
Arch. Hal. di Anat. x, p. 545, 1911; also E. le Breton and G. Schaeffer, Variations
hiuchimiqiies du rapport nucleo-plasmatique, Strasburg, 1923.

t Arch.f. Entw. Mech. iv, 1898, pp. 75, 247.

X E. G. Conklin, Cell-size and nuclear size, Journ. Exp. Zool. xii, pp. 1-98,
1912; Body-size and cell-size, Journ. of Morphol. xxiii, pp. 159-188, 1912. Cf.
M. Popoff, Ueber die Zellgrosse, Arch.f. Zellforschung, iii, 1909.

§ Thus the fibres of the crystalline lens are of the same size in large and small
dogs, Rabl, Z.f. w. Z. lxvii, 1899. Cf. {int. al.) Pearson, On the size of the blood-
corpuscles in Rana, Biometrika, vi, p. 403, 1909. Dr Thomas Young caught sight
of the phenomenon early in last century: "The solid particles of the blood do not
by any means vary in magnitude in the same ratio with the bulk of the animal,"
Natural Philosophy, ed. 1845, p. 466; and Leeuwenhoek and Stephen Hales
were aware of it nearly two hundred years before. Leeuwenhoek indeed had
a very good idea of the size of a human blood-corpuscle, and was in the habit of
using its diameter — about 1/3000 of an inch — as a standard of comparison. But
though the blood-corpuscles shew no relation of magnitude to the size of the
animal, they are related without doubt to its activity; for the corpuscles in the

OF THE SIZE OF CELLS

61

the leaf-cells are found to be of the same size in an ordinary water-
lily, in the great Victoria regia, and in the still hiiger leaf, nearly
3 metres long, of Euryale ferox in Japan*. Driesch has laid par-
ticular stress upon this principle of a "fixed cell-size," which has,
however, its own limitations and exceptions. Among these excep-
tions, or apparent exceptions, are the giant frond-like cell of a
Caulerpa or the great undivided plasmodium of a Myxomycete.
The flattening of the one and the branching of the other serve (or
help) to increase the ratio of surface to content, the nuclei tend to
multiply, and streaming currents keep the interior and exterior of
the mass in touch with one another.

j^

Rabbit

Man Dog

Fig. 3. Motor ganglion-cells, from the cervical spinal cord.
From Minot, after Irving Hardesty.

We get a good and even a famiUar illustration of the principle
of size-hmitation in comparing the brain-cells or ganghon-cells,
whether of the lower or of the higher animals f . In Fig. 3 we shew
certain identical nerve-cells from various mammals, from mouse to
elephant, all drawn to the same scale of magnification ; and we see
that they are all of much the same order of magnitude. The nerve-
cell of the elephant is about twice that of the mouse in linear

sluggish Amphibia are much the largest known to us, while the smallest are found
among the deer and other agile and speedy animals (cf. Gulliver, P.Z.S. 1875,
p. 474, etc.). This correlation is explained by the surface condensation or
adsorption of oxygen in the blood-corpuscles, a process greatly facilitated and
intensified by the increase of surface due to their minuteness.

* Okada and Yomosuke, in Sci. Rep. Tohoku Univ. iii, pp. 271-278, 1928.

t Cf. P. Enriques, La forma eome funzione della grandezza : Ricerche sui gangli
nervosi degli invertebrati, Arch. f. Entw. Mech. xxv, p. 655, 1907-8.

62 ON MAGNITUDE [ch.

dimensions, and therefore about eight times greater in volume or
in mass. But making due allowance for difference of shape, the
linear dimensions of the elephant are to those of the mouse as not
less than one to fifty ; and the bulk of the larger animal is something
like 125,000 times that of the less. It follows, if the size of the
nerve-cells are as eight to one, that, in corresponding parts of the
nervous system, there are more than 15,000 times as many individual
cells in one animal as in the other. In short we may (with Enriques)
lay it down as a general law that among animals, large or small, the
gangUon-cells vary in size within narrow limits; and that, amidst
all the great variety of structure observed in the nervous system
of different classes of animals, it is always found that the smaller
species have simpler gangha than the larger, that is to say ganglia
containing a smaller number of cellular elements*. The bepiing of
such facts as this upon the cell-theory in general is not to be dis-
regarded; and the warning is especially clear against exaggerated
attempts to correlate physiological processes with the visible
mechanism of associated cells, rather than with the system of
energies, or the field of force, which is associated with them. For
the life of the body is more than the swm of the properties of the
cells of which it is composed: as Goethe said, "Das Lebendige ist
zwar in Elemente zerlegt,* aber man kann es aus diesen nicht wieder
zusammenstellen und beleben."

Among certain microscopic organisms such as the Rotifera (which
have the least average size and the narrowest range of size of all
the Metazoa), we are still more palpably struck by the small number
of cells which go to constitute a usually complex organ, such as
kidney, stomach or ovary ; we can sometimes number them in a few

* While the difference in cell-volume is vastly less than that between the
volumes, and very much less also than that between the surfaces, of the respective
animals, yet there is a certain difference ; and this it has been attempted to correlate
with the need for each cell in the many-celled ganglion of the larger animal to
possess a more complex "exchange-system" of branches, for intercommunication
with its more numerous neighbours. Another explanation is based on the fact
that, while such cells as continue to divide throughout life tend to uniformity of
size in all mammals, those which do not do so, and in particular the ganglion cells,
continue to grow, and their size becomes, therefore, a function of the duration of
life. Cf. G. Levi, Studii sulla grandezza delle cellule. Arch. Jtal. di Anat. e di
Embrioloq. v, p. 291, 1906; cf. also A. Berezowski, Studien liber die Zellgrosse,
Arch. f. Zellforsch. v, pp. 375-384, 1910.

II] OF THE LEAST OF ORGANISMS 63

units, in place of the many thousands which make up such an organ
in larger, if not alwayc higher, animals. We have already spoken
of the Fairy-flies, a few score of which would hardly weigh down
one of the larger rotifers, and a hundred thousand would weigh less
than one honey-bee. Their form is complex and their httle bodies
exquisitely beautiful; but I feel sure that their cells are few, and
their organs of great histological simplicity. These considerations
help, I think, to shew that, however important and advantageous
the subdivision of the tissues into cells may be from the construc-
tional, or from the dynamic, point of view, the phenomenon has
less fundamental importance than was once, and is often still,
assigned to it.

Just as Sachs shewed there was a hmit to the amount of cytoplasm
which could gather round a nucleus, so Boveri has demonstrated
that the nucleus itself has its own hmitations of size, and that, in
cell-division after fertilisation, each new nucleus has the same size
as its parent nucleus*; we may nowadays transfer the statement
to the chromosomes. It may be that a bacterium lacks a nucleus
for the simple reason that it is too small to hold one, and that the
same is true of such small plants as the Cyanophyceae, or blue-green
algae. Even a chromatophore with its "pyrenoids" seems to be
impossible below a certain sizej.

Always then, there are reasons, partly physiological but in large
part purely physical, which define or regulate the magnitude of the
organism or the cell. And as we have already found definite
Hmitations to the increase in magnitude of an organism, let us now
enquire whether there be not also a lower limit below which the
very existence of an organism becomes impossible.

* Boveri, Zellen.studien, V: Ueber die Abhangigkeit der Kerngrosse und
Zellenzahl von der Chromosomenzahl der Ausgangszellen. Jena, 1905. Cf. also
{int. al.) H. Voss, Kerngrossenverhaltnisse in der Leber etc., Ztschr.f. Zellforschung,
VII, pp. 187-200, 1928.

t The size of the nucleus may be affected, even determined, by the number of
chromosomes it contains. There are giant race* of Oenothera, Primula and Solanum
whose cell-nuclei contain twice the normal number of chromosomes, and a dwarf
race of a little freshwater crustacean, Cyclops, has half the usual number. The
cytoplasm in turn varies with the amount of nuclear matter, the whole cell is
unusually large or unusually small; and in these exceptional cases we see a direct
relation between the size of the organism and the size of the cell. Cf. {int. al.)
R. P. Gregory, Proc. Camb. Phil Soc. xv, pp. 239-246, 1909; F. Keeble, Journ.
of Genetics, ii, pp. 163-188, 1912. •

64 ON MAGNITUDE [ch.

A bacillus of ordinary size is, say, 1 /x in length. The length (or
height) of a man is about a million and three-quarter times as great,
i.e. 1-75 metres, or 1-75 x 10^ /x; and the mass of the man is in the
neighbourhood of 5 x 10^® (five million, million, milHon) times
greater than that of the bacillus. If we ask whether there may not
exist organisms as much less than the bacillus as the bacillus is less
than the man, it is easy to reply that this is quite impossible, for we
are rapidly approaching a point where the question of molecular
dimensions, and of the ultimate divisibility of matter, obtrudes
itself as a crucial factor in the case. Clerk Maxwell dealt with this
matter seventy years ago, in his celebrated article Atom"^. KoUi
(or Colley), a Russian chemist, declared in 1893 that the head of a
spermatozoon could hold no more than a few protein molecules ; and
Errera, ten years later, discussed the same topic with great ingenuity f.
But it needs no elaborate calculation to convince us that the smaller
bacteria or micrococci nearly approach the smallest magnitudes
which we can conceive to have an organised structure. A few small
bacteria are the smallest of visible organisms, and a minute species
associated with influenza, B. pneumosinter, is said to be the least
of them all. Its size is of the order of 0-1 /x, or rather less; and
here we are in close touch with the utmost limits of microscopic
vision, for the wave-lengths of visible light run only from about
400 to 700 m/x. The largest of the bacteria, B. megatherium, larger
than the well-known B. anthracis of splenic fever, has much the
same proportion to the least as an elephant to a guinea-pig {.

Size of body is no mere accident. Man, respiring as he does,
cannot be as small as an insect, nor vice versa; only now and then,
as in the Goliath beetle, do the sizes of mouse and beetle meet and
overlap. The descending scale of mammals stops short at a weight
of about 5 grams, that of beetles at a length of about half a milli-
metre, and every group of animals has its upper and its lower
limitations of size. So, not far from the lower limit of our vision,
does the long series of bacteria come to an end. There remain still
smaller particles which the ultra-microscope in part reveals; and

* Encyclopaedia Britannica, 9th edition, 1875.

f Leo Errera, Siir la limite de la petitesse des organismes, BvlL Soc. Roy. des
Sc. me'd. et nat. de Bruxelles, 1903; Recueil d'osnvres {Physiologie gen^rale), p. 325.

I Cf. A. E. Boycott, The transition from live to dead, Proc. R. Soc. of Medicine,
XXII {Pathology), pp. 55-69, 1928.

II] OF MOLECULAR MAGNITUDES 65

here or hereabouts are said to come the so-called viruses or "filter-
passers," brought within our ken by the maladies, such as hydro-
phobia, or foot-and-mouth disease, or the mosaic diseases of tobacco
and potato, to which they give rise. These minute particles, of the
order of one-tenth the diameter of our smallest bacteria, have no
diffusible contents, no included water — whereby they differ from
every living thing. They appear to be inert colloidal (or even
crystalloid) aggregates of a nucleo-protein, of perhaps ten times the
diameter of an ordinary protein-molecule, and not much larger than
the giant molecules of haemoglobin or haemocyanin *.

Bejerinck called such a virus a contagium vivum; "infective
nucleo-protein" is a newer name. We have stepped down, by a
single step, from Hving to non-Hving things, from bacterial dimen-
sions to the molecular magnitudes of protein chemistry. And we
begin to suspect that the virus-diseases are not due to an "organism,
capable of physiological reproduction and multiphcation, but to a
mere specific chemical substance, capable of catalysing pre-existing
materials and thereby producing more and more molecules hke
itself. The spread of the virus in a plant would then be a mere
autocatalysis, not involving the transport of matter, but only a
progressive change of state in substances already there f."

But, after all, a simple tabulation is all we need to shew how
nearly the least of organisms approach to molecular magnitudes.
The same table will suffice to shew how each main group of animals
has its mean and characteristic size, and a range on either side,
sometimes greater and sometimes less.

Our table of magnitudes is no mere catalogue of isolated facts,
but goes deep into the relation between the creature and its world.
A certain range, and a- narrow one, contains mouse and elephant,
and all whose business it is to walk and run ; this is our own world,

* Cf. Svedberg, Journ. Am. Chem. Soc. XLviir, p. 30, 1926. According to the
Foot-and-Mouth Disease Research Committee {oth Report, 1937), the foot-and-
mouth virus has a diameter, determined by graded filters, of 8-12m/i; while
Kenneth Smith and W. D. MacClement {Proc. R.S. (B), cxxv, p. 296, 1938) calculate
for certain others a diameter of no more than 4m^, or less than a molecule of
haemocyanin.

t H. H. Dixon, Croonian lecture on the transport of substances in plants,
Proc. R.S. (B), vol. cxxv, pp. 22, 23, 1938.

66

ON MAGNITUDE

[CH.

with whose dimensions our hves, our hmbs, our senses are in tune.
The great whales grow out of this range by throwing the burden
of their bulk upon the waiters; the dinosaurs wallowed in the swamp,
and the hippopotamus, the sea-elephant and Steller's great sea-cow
pass or passed their hves in the rivers or the sea. The things which

Linear dimensions of organisms, and other objects

cm.

(10,000 km.)

10'

A quadrant of the earth's circumference

(1000 km.)

10«
105
10*

Orkney to Land's End
Mount Everest

(km.)

103

102

Giant trees: Sequoia
Large whale
Basking shark

101

Elephant; ostrich; man

(metre)

10"
10-1

Dog; rat; eagle

Small birds and mammals; large insects

(cm.)

10-2

Small insects; minute fish

(mm.)

10-3

Minute insects

10-*

Protozoa; pollen -grains

- Cells

10-5

Large bacteria; human blood-corpuscles

(micron, /z,)

io-«

10-'

Minute bacteria

Limit of microscopic vision

io-«

Viruses, or filter-passers r. n j *• i
Giant albuminoids, casein, etc.f ^^^^^^ P^^^^^^^^
Starch-molecule

(m/ii)

io-»

Water-molecule

(Angstrom unit)

10-10

fly are smaller than the things which walk and run ; the flying birds
are never as large as the larger mammals, the lesser birds and
mammals are much of a muchness, but insects come down a step
in the scale and more. The lessening influence of gravity facilitates
flight, but makes it less easy to walk and run; first claws, then
hooks and suckers and glandular hairs help to secure a foothold,

II] OF SCALES OF MAGNITUDE 67

until to creep upon wall or ceiling becomes as easy as to walk upon
the ground. Fishes, by evading gravity, increase their range of
magnitude both above and below that of terrestrial animals. Smaller
than all these, passing out of our range of vision and going down to
the least dimensions of hving things, are protozoa, rotifers, spores,
pollen-grains* and bacteria. All save the largest of these float
rather than swim; they are buoyed up by air or water, and fall
(as Stokes's law explains) with exceeding slowness.

There is a certain narrow range of magnitudes where (as we have
partly said) gravity and surface tension become comparable forces,
nicely balanced with one another. Here a population of small
plants and animals not only dwell in the surface waters but are
bound to the surface film itself — the whirligig beetles and pond-
skaters, the larvae of gnat and mosquito, the duckweeds (Lemna),
the tiny Wolffia, and Azolla; even in mid-ocean, one small insect
(Halobates) retains this singular habitat. It would be a long story
to tell the various ways in which surface-tension is thus taken full
advantage of. Gravitation not only Hmits the magnitude but
controls the form of things. With the help of gravity the quadruped
has its back and its belly, and its limbs upon the ground ; its freedom
of motion in a plane perpendicular to gravitational force ; its sense
of fore-and-aft, its head and tail, its bilateral symmetry. Gravitation
influences both our bodies and our minds. We owe to it our sense
of the vertical, our knowledge of up-and-down; our conception of
the horizontal plane on which we stand, and our discovery of two
axes therein, related to the vertical as to one another; it was gravity
which taught us to think of three-dimensional space. Our archi-
tecture is controlled by gravity, but gravity has less influence over
the architecture of the bee ; a bee might be excused, might even be
commended, if it referred space to four dimensions instead of three ! t
The plant has its root and its stem", but about this vertical or

* Pollen-grains, like protozoa, have a considerable range of magnitude. The
largest, such as those of the pumpkin, are about 200/x in diameter; these have to
be carried by insects, for they are above the level of Stokes's law, and no longer
float upon the air. The smallest pollen-grains, such as those of the forget-me-not,
are about 4^ /x in diameter (Wodehouse).

f Corresponding, that is to say, to the four axes which, meeting in a pfoint, make
co-equal angles (the so-called tetrahedral angles) one with another, as do the basal
angles of the honeycomb. (See below, chap, vu.)

68 ON MAGNITUDE [ch.

gravitational axis its radiate symmetry remains, undisturbed by
directional polarity, save for the sun. Among animals, radiate
symmetry is confined to creatures of no great size ; and some form
or degree of spherical symmetry becomes the rule in the small world
of the protozoon — unless gravity resume its sway through the added
burden of a shell. The creatures which swim, walk or run, fly,
creep or float are, so to speak, inhabitants and natural proprietors
of as many distinct and all but separate worlds. Humming-bird
and hawkmoth may, once in a way, be co-tenants of the same
world; but for the most part the mammal, the bird, the fish, the
insect and the small life of the sea, not only have their zoological
distinctions, but each has a physical universe of its own. The
world of bacteria is yet another world again, and so is the world of
colloids ; but through these small Lilliputs we pass outside the range
of hving things.

What we call mechanical principles apply to the magnitudes
among which we are at home; but lesser worlds are governed by
other and appropriate physical laws, of capillarity, adsorption and
electric charge. There are other worlds at the far other end of the
scale, in the uttermost depths of space, whose vast magnitudes lie
within a narrow range. When the globular star-clusters are plotted
on a curve, apparent diameter against estimated distance, the
curve is a fair approximation to a rectangular hyperbola; which
means that, to the same rough approximation, the actual diameter
is identical in them all*.

It is a remarkable thing, worth pausing to reflect on, that we can
pass so easily and in a dozen fines from molecular magnitudes f to
the dimensions of a Sequoia or a whale. Addition and subtraction,
the old arithmetic of the Egyptians, are not powerful enough for
such an operation; but the story of the grains of wheat upon the
chessboard shewed the way, and Archimedes and Napier elaborated

* See Harlow Shapley and A. B. Sayer, The angular diameters of globular
clusters, Proc. Nat. Acad, of Sci. xxi, pp. 593-597, 1935. The same is approxi-
mately true of the spiral nebulae also.

t We may call (after Siedentopf and Zsigmondi) the smallest visible particles
microns, such for instance as small bacteria, or the fine particles of gum-mastich
in suspension, measuring 0-5 to 1-0/x; sub-microns are those revealed by the ultra-
microscope, such as particles of colloid gold (2-15m/Lt), or starch-moleculea (5m/x);
amicrons, under Im^, are not perceptible by either method. A water-molecule
measures, probably, about 0-1 m/i.

II] OF THIN FILMS 69

the arithmetic of multipHcation. So passing up and down by easy-
steps, as Archimedes did when he numbered the sands of the sea,
we compare the magnitudes of the great beasts and the small, of
che atoms of which they are made, and of the world in which they
dwell*.

While considerations based on the chemical composition of the
organism have taught us that there must be a definite lower hmit
to its magnitude, other considerations of a purely physical kind lead
us to the same conclusion. For our discussion of the principle of
similitude has already taught us that long before we reach these
all but infinitesimal magnitudes the dwindling organism will have
experienced great changes in all its physical relations, and must at
length arrive at conditions surely incompatible with life, or what we
understand as life, in its ordinary development and manifestation.

We are told, for instance, that the powerful force of surface-tension,
or capillarity, begins to act within a range of about 1/500,000 of an
inch, or say 0-05 />t. A soap film, or a film of oil on water, may be
attenuated to far less magnitudes than this; the black spots on a
soap bubble are known, by various concordant methods of measure-
ment, to be only about 6x 10~' cm., or about 6m/x thick, and Lord
Rayleigh and M. Devaux have obtained films of oil of 2mjLt, or even
1 m/x in thickness. But while it is possible for a fluid film to exist
of these molecular dimensions, it is certain that long before we
reach these magnitudes there arise conditions of which we have
little knowledge, and which it is not easy to imagine. A bacillus
lives in a world, or on the borders of a world, far other than our
own, and preconceptions drawn from our experience are not valid
there. Even among inorganic, non-living bodies, there comes a
certain grade of minuteness at which the ordinary properties become
modified. For instance, while under ordinary circumstances crystal-
lisation starts in a solution about a minute sohd fragment or crystal

* Observe that, following a common custom, we have only used a logarithmic
scale for the round numbers representing powers of ten, leaving the interspaces
between these to be filled up, if at all, by ordinary numbers. There is nothing
to prevent us from using fractional indices, if we please, throughout, and calling
a blood-corpuscle, for instance, 10~^"^ cm. in diameter, a man lO^'^^ cm. high, or
Sibbald's Rorqual lOi"*^ metres long. This method, implicit in that of Napier of
Merchiston, was first set forth by Wallis, in his Arithmetica infinitorum.

70 ON MAGNITUDE [ch.

of the salt, Ostwald has shewn that we may have particles so minute
that they fail to serve as a nucleus for crystalUsation — which is as
much as to say that they are too small to have the form and pro-
perties of a "crystal." And again, in his thin oil-films, Lord
Rayleigh noted the striking change of physical properties which
ensues when the film becomes attenuated to one, or something less
than one, close-packed layer of molecules, and when, in short, it no
longer has the properties of matter in mass.

These attenuated films are now known to be "monomolecular," the
long-chain molecules of the fatty acids standing close-packed, like the cells
of a honeycomb, and the film being just as thick as the molecules are long.
A recent determination makes the several molecules of oleic, palmitic and
stearic acids measure 10-4, 14-1 and 15- 1 cm. in length, and in breadth 7-4,
6-0 and 5-5 cm., all by 10~^: in good agreement with Lord Rayleigh and
Devaux's lowest estimates (F. J. Hill, Phil. Mag. 1929, pp. 940-946). But
it has since been shewn that in aliphatic substances the long-chain molecules
are not erect, but inclined to the plane of the film ; that the zig-zag constitution
of the molecules permits them to interlock, so giving the film increased
stability ; and that the interlock may be by means of a first or second zig-zag,
the measured area of the film corresponding precisely to these two dimorphic
arrangements. (Cf. C. G. Lyons and E. K. Rideal, Proc. R.S. (A), cxxvin,
pp. 468-473, 1930.) The film may be lifted on to a polished surface of metal,
or even on a sheet of paper, and one monomolecular layer so added to another;
even the complex protein molecule can be unfolded to form a film one amino-
acid molecule thick. The whole subject of monomolecular layers, the nature
of the film, whether condensed, expanded or gaseous, its astonishing sensitive-
ness to the least impurities, apd the manner of spreading of the one liquid
over the other, has become of great interest and importance through the work
of Irving Langmuir, Devaux, N. K. Adam and others, and throws new light
on the whole subject of molecular magnitudes*.

The surface-tension of a drop (as Laplace" conceived it) is the
cumulative effect, the statistical average, of countless molecular
attractions, but we are now entering on dimensions where the
molecules are fewf. The free surface-energy of a body begins to
vary with the radius, when that radius is of an order comparable
to inter-molecular distances; and the whole expression for such
energy tends to vanish away when the radius of the drop or particle
is less than O-Olfx, or lOm^it. The quahties and properties of our

* Cf. (int. al.) Adam, Physics and Chemistry of Surfaces, 1930; Irving Langmuir,
Proc. R.S. (A), CLXX, 1939.

t See a very interesting paper by Fred Vies, Introduction a la physique bac-
terienne, Revue Sclent. 11 juin 1921. Cf. also N. Rashevsky, Zur Theorie d.
spontanen Teilung von mikroskopischen Tropfen, Ztschr.f. Physik, xlvi, p. 578, 1928.

II] OF MINUTE MAGNITUDES 71

particle suffer an abrupt change here; what then can we attribute,
in the way of properties, to a corpuscle or organism as small or
smaller than, say, 0-05 or 0-03 /x? It must, in all probability, be a
homogeneous structureless body, composed of a very small number
of albumenoid or other molecules. Its vital properties and functions
must be extremely limited; its specific outward characters, even if
we could see it, must be nil; its osmotic pressure and exchanges
must be anomalous, and under molecular bombardment they may
be rudely disturbed; its properties can be Httle more than those of
an ion-laden corpuscle, enabling it to perform this or that specific
chemical reaction, to effect this or that disturbing influence, or
produce this or that pathogenic effect. Had it sensation, its ex-
periences would be strange indeed; for if it could feel, it would regard
a fall in temperature as a movement of the molecules around, and
if it could see it would be surrounded with light of many shifting
colours, like a room filled with rainbows.

The dimensions of a cilium are of such an order that its substance
is mostly, if not all, under the pecuHar conditions of a surface-layer,
and surface-energy is bound to play a leading part in cihary action.
A cilium or flagellum is (as it seems to me) a portion of matter in
a state sui generis, with properties of its own, just as the film and the
jet have theirs. And just as Savart and Plateau have told us about
jets and films, so will the physicist some day explain the properties
of the cilium and flagellum. It is certain that we shall never
understand these remarkable structures so long as we magnify
them to another scale, and forget that new and pecuhar physical
properties are associated with the scale to which they belong*.

As Clerk Maxwell put it, "molecular science sets us face to face
with physiological theories. It forbids the physiologist to imagine
that structural details of infinitely small dimensions (such as Leibniz
assumed, one within another, ad infinitum) can furnish an explana-
tion of the infinite variety which exists in the properties and functions
of the most minute organisms." And for this reason Maxwell
reprobates, with not undue severity, those advocates of pangenesis

* The cilia on the gills of bivalve molluscs are of exceptional size, measuring
from say 20 to 120^ long. They are thin triangular plates, rather than filaments;
they are from 4 to lO/x broad at the base, but less than 1/x thick. Cf. D. Atkins.
Q.J. M.S., 1938, and other papers.

72 ON MAGNITUDE [ch.

and similar theories of heredity, who "would place a whole world
of wonders within a body so small and so devoid of visible structure
as a germ." But indeed it scarcely needed Maxwell's criticism to
shew forth the immense physical difficulties of Darwin's theory of
pangenesis: which, after all, is as old as Democritus, and is no other
than that Promethean particula undique desecta of which we have
read, and at which we have smiled, in our Horace.

There are many other ways in which, when we make a long
excursion into space, we find our ordinary rules of physical behaviour
upset. A very familiar case, analysed by Stokes, is that the
viscosity of the surrounding medium has a relatively powerful effect
upon bodies below a certain size. A droplet of water, a thousandth
of an inch (25 ju,) in diameter, cannot fall in still air quicker than
about an inch and a half per second; as its size decreases, its
resistance varies as the radius, not (as with larger bodies) as the
surface; and its "critical" or terminal velocity varies as the
square of the radius, or as the surface of the drop. A minute
drop in a misty cloud may be one-tenth that size, and will fall a
hundred times slower, say an inch a minute; and one again a tenth
of this diameter (say 0-25 />t, or about twice as big as a small micro-
coccus) will scarcely fall an inch in two hours*. Not only do
dust-particles, spores f and bacteria fall, by reason of this principle,
very slowly through the air, but all minute bodies meet with great
proportionate resistance to their movements through a fluid. In
salt water they have the added influence of a larger coefficient of
friction than in fresh J ; and even such comparatively large organisms
as the diatoms and the foraminifera, laden though they are with a
heavy shell of flint or lime, seem to be poised in the waters of the
ocean, and fall with exceeding slowness.

* The resistance depends on the radius of the particle, the viscosity, and the
rate of fall ( V) ; the eifective weight by which this resistance is to be overcome
depends on gravity, on the density of the particle compared with that of the
medium, and on the mass, which varies as r^. Resistance =A;rF, and effective
weight = )fcV; when these two equal one another we have the critical or terminal
velocity, and Vccr^.

t A. H. R. BuUer found the spores of a fungus (CoUybia), measuring 5x3/i,
to fall at the rate of half a millimetre per second, or rather more than an inch
a minute; Studies on Fungi, 1909.

X Cf. W. Krause, Biol. Centralbl. i, p. 578, 1881; Fliigel, Meteorol Ztschr. 1881,
p. 321.

II] OF STOKES'S LAW 73

When we talk of one thing touching another, there may yet be
a distance between, not only measurable but even large compared
with the magnitudes we have been considering. Two polished
plates of glass or steel resting on one another are still about 4/x
apart — the average size of the smallest dust; and when all dust-
particles are sedulously excluded, the one plate sinks slowly down
to within O-S/jl of the other, an apparent separation to be accounted
for by minute irregularities of the polished surfaces*.

The Brownian movement has also to be reckoned with — that
remarkable phenomenon studied more than a century ago by Robert
Brown f, Humboldt's /ac?7e princeps botanicorum, and discoverer of
the nucleus of the cell J. It is the chief of those fundamental
phenomena which the biologists have contributed, or helped to
contribute, to the science of physics.

The quivering motion, accompanied by rotation and even by
translation, manifested by the fine granular particle issuing from a
crushed pollen-grain, and which Brown proved to have no vital
significance but to be manifested by all minute particles whatsoever,
was for many years unexplained. Thirty years and more after Brown
wrote, it was said to be "due, either directly to some calorical
changes continually taking place in the fluid, or to some obscure
chemical action between the solid particles and the fluid which is
indirectly promoted by heat§." Soon after these words were

* Cf. Hardy and Nottage, Proc. R.S. (A), cxxviii, p. 209, 1928; Baston and
Bowden, ibid, cxxxiv, p. 404, 1931.

t A Brief Description of Microscopical Observations. . .on the Particles contained
in the Pollen of Plants; and on the General Existence of Active Molecules in Organic
and Inorganic Bodies, London, 1828. See also Edinb. Netv Philosoph. Journ. v,
p. 358, 1828; Edinb. Journ. of Science, i, p. 314, 1829; Ann. Sc. Nat. xiv, pp. 341-
362, 1828; etc. The Brownian movement was hailed by some as supporting
Leibniz's theory of Monads, a theory once so deeply rooted and so widely believed
that even under Schwann's cell-theory Johannes Miiller and Henle spoke of
the cells as "organische Monaden"; cf. Emit du Bois Reymond, Leibnizische
Gedanken in der neueren Naturwissenschaft, Monatsber. d. k. Akad. Wiss., Berlin,
1870.

J The "nucleus" was first seen in the epidermis of Orchids; but "this areola,
or nucleus of the cell as perhaps it might be termed, is not confined to the
epidermis," etc. See his paper on Fecundation in Orchideae and Asclepiadae,
Trans. Linn. Soc. xvi, 1829-33, also Proc. Linn. Soc. March 30, 1832.

§ Carpenter, The Microscope, edit. 1862, p. 185.

74 ON MAGNITUDE [ch.

written it was ascribed by Christian Wiener * to molecular move-
ments within the fluid, and was hailed as visible proof of the
atomistic (or molecular) constitution of the same. We now know
that it is indeed due to the impact or bombardment of molecules
upon a body so small that these impacts do not average out, for
the moment, to approximate equality on all sides f. The movement
becomes manifest with particles of somewhere about 20 /x, and is
better displayed by those of about 10 /x, and especially well by
certain colloid suspensions or emulsions whose particles are just
below 1/Lt in diameter {. The bombardment causes our particles to
behave just hke molecules of unusual size, and this behaviour is
manifested in several ways§. Firstly, we have the quivering
movement of the particles; secondly, their movement backwards
and forwards, in short, straight disjointed paths; thirdly, the
particles rotate, and do so the more rapidly the smaller they are:
and by theory, confirmed by observation, it is found that particles
of IjjL in diameter rotate on an average through 100° a second,
while particles of 13/x turn through only 14° a minute. Lastly, the
very curious result appears, that in a layer of fluid the particles are
not evenly distributed, nor do they ever fall under the influence of
gravity to the bottom. For here gravity and the Brownian move-
ment are rival powers, striving for equilibrium; just as gravity is
opposed in the atmosphere by the proper motion of the gaseous
molecules. And just as equihbrium is attained in the atmosphere
when the molecules are so distributed that the density (and therefore
the number of molecules per unit volume) falls oif in geometrical

* In Poggendorffs Annalen, cxviii, pp. 79-94, 1863. For an account of this
remarkable man, see Naturmssensehaften, xv, 1927; cf. also Sigraund Exner,
Ueber Brown's Molecularbewegung, Sitzungsher. kk. Akad. Wien, lvi, p. 116, 1867.

t Perrin, Les preuves de la realite moleculaire, Ann. de Physique, xvii, p. 549,
1905; XIX, p. 571, 1906. The actual molecular collisions are unimaginably
frequent; we see only the residual fluctuations.

J Wiener was struck by the fact that the phenomenon becomes conspicuous
just when the size of the particles becomes comparable to that of a wave-length
of light.

§ For a full, but still elementary, account, see J. Perrin, Les Atomes; cf. also
Th. Svedberg, Die Existenz der Molekiile, 1912; R. A. Millikan, The Electron,
1917, etc. The modern literature of the Brownian movement (by Einstein, Perrin,
de Broglie, Smoluchowski and Millikan) is very large, chiefly owing to the value
which the phenomenon is shewn to have in determining the size of the atom or
the charge on an electron, and of giving, as Ostwald said, experimental proof of
the atomic theory.

II] OF THE BROWNIAN MOVEMENT 75

progression as we ascend to higher and higher layers, so is it with
our particles within the narrow Umits of the little portion of fluid
under our microscope.

It is only in regard to particles of the simplest form that these
phenomena have been theoretically investigated*, and we may take
it as certain that more complex particles, such as the twisted body
of a Spirillum, would shew other and still more comphcated mani-
festations. It is at least clear that, just as the early microscopists
in the days before Robert Brown never doubted but that these
phenomena were purely vital, so we also may still be apt to confuse,
in certain cases, the one phenomenon with the other. We cannot,
indeed, without the most careful scrutiny, decide whether the
movements of our minutest organisms are intrinsically "vital" (in
the sense of being beyond a physical mechanism, or working model)
or not. For example, Schaudinn has suggested that the undulating
movements of Spirochaete pallida must be due to the presence of a
minute, unseen, "undulating membrane"; and Doflein says of the
same species that "sie verharrt oft mit eigenthiimlich zitternden
Bewegungen zu einem Orte." Both movements, the trembling or
quivering movement described by Doflein, and the undulating or
rotating movement described by Schaudinn, are just such as may
be easily and naturally interpreted as part and parcel of the Brownian
phenomenon.

While the Brownian movement may thus simulate in a deceptive
way the active movements of an organism, the reverse statement
also to a certain extent holds good. One sometimes Hes awake of
a summer's morning watching the flies as they dance under the
ceiling. It is a very remarkable dance. The dancers do not whirl or
gyrate, either in company or alone; but they advance and retire;
they seem to jostle and rebound; between the rebounds they dart
hither or thither in short straight snatches of hurried flight, and
turn again sharply in a new rebound at the end of each little rushf.

* Cf. R. Gans, Wie fallen Stabe und Scheiben in einer reibenden Fliissigkeit?
Miinchener Bericht, 1911, p. 191; K. Przibram, Ueber die Brown'sche Bewegung
nicht kugelformiger Teilchen, Wiener Bericht, 1912, p. 2339; 1913, pp. 1895-1912.

t As Clerk Maxwell put it to the British Association at Bradford in 1873, "We
cannot do better than observe a swarm of bees, where every individual bee is
flying furiously, first in one direction and then in another, while the swarm as
a whole is either at rest or sails slowly through the air."

76 ON MAGNITUDE [ch.

Their motions are erratic, independent of one another, and
devoid of common purpose*. This is nothing else than a
vastly magnified picture, or simulacrum, of the Brownian move-
ment; the parallel between the two cases lies in their complete
irregularity, but this in itself implies a close resemblance. One
might see the same thing in a crowded market-place, always provided
that the busthng crowd had no business whatsoever. In like
manner Lucretius, and Epicurus before him, watched the dust-motes
quivering in the beam, and saw in them a mimic representation,
rei simulacrum et imago, of the eternal motions of the atoms. Again
the same phenomenon may be witnessed under the microscope, in
a drop of water swarming with Paramoecia or such-Hke Infusoria;
and here the analogy has been put to a numerical test. Following
with a pencil the track of each little swimmer, and dotting its place
every few seconds (to the beat of a metronome), Karl Przibram
found that the mean successive distances from a common base-hne
obeyed with great exactitude the "Einstein formula," that is to
say the particular form of the "law of chance" which is apphcable
to the case of the Brownian movement f. The phenomenon is (of
course) merely analogous, and by no means identical with the
Brownian movement; for the range of motion of the little active
organisms, whether they be gnats or infusoria, is vastly greater than
that of the minute particles which are passive under bombardment ;
nevertheless Przibram is inclined to think that even his compara-
tively large infusoria are small enough for the molecular bombard-
ment to be a stimulus, even though not the actual cause, of their
irregular and interrupted movements {.

* Nevertheless there may be a certain amount of bias or direction in these
seemingly random divagations: cf. J. Brownlee, Proc. R.S.E. xxxi, p. 262,
1910-11; F. H. Edgeworth, Metron, i, p. 75, 1920; Lotka, Elem. of Physical
Biology, 1925, p. 344.

t That is to say, the mean square of the displacements of a particle, in any
direction, is proportional to the interval of time. Cf. K. Przibram, Ueber die
ungeordnete Bewegung niederer Tiere, Pfliigefs Archiv, cliii, pp. 401-405, 1913;
Arch. f. Entw. Mech. xliii, pp. 20-27, 1917.

X All that is actually proven is that "pure chance" has governed the movements
of the little organism. Przibram has made the analogous observation that
infusoria, when not too crowded together, spread or diffuse through an aperture
from one vessel to another at a rate very closely comparable to the ordinary laws
of molecular diffusion.

II] OF THE EFFECTS OF SCALE 77

George Johnstone Stoney, the remarkable man to whom we owe
the name and concept of the electron, went further than this; for
he supposed that molecular bombardment might be the source of
the life-energy of the bacteria. He conceived the swifter moving
molecules to dive deep into the minute body of the organism, and
this in turn to be able to make use of these importations of energy*.

We draw near the end of this discussion. We found, to begin
with, that "scale" had a marked eifect on physical phenomena, and
that increase or diminution of magnitude migl^t mean a complete
change of statical or dynamical equiUbrium. In the end we begin
to see that there are discontinuities in the scale, defining phases in
which different forces predominate and different conditions prevail.
Life has a range of magnitude narrow indeed compared to that with
which physical science deals ; but it is wide enough to include three
such discrepant conditions as those in which a man, an insect and
a bacillus have their being and play their several roles. Man is
ruled by gravitation, and rests on mother earth. A water-beetle
finds the surface of a pool a matter of Ufe and death, a perilous
entanglement or an indispensable support. In a third world,
where the bacillus Hves, gravitation is forgotten, and the viscosity
of the Hquid, the resistance defined by Stokes's law, the molecular
shocks of the Brownian movement, doubtless also the electric
charges of the ionised medium, make up the physical environment
and have their potent and immediate influence on the organism.
The predominant factors are no longer those of our scale ; we have
come to the edge of a world of which we have no experience, and
where all our preconceptions must be recast.

* Phil. Mag. April 1890.

CHAPTER III,

THE RATE OF GROWTH

When we study magnitude by itself, apart from the gradual
changes to which it may be subject, we are deahng with a something
which may be adequately represented by a number, or by means
of a Hne of definite length ; it is what mathematicians call a scalar
phenomenon. When we introduce the conception of change of
magnitude, of magnitude which varies as we pass from one point
to another in space, or from one instant to another in time, our
phenomenon becomes capable of representation by means of a line
of which we define both the length and the direction; it is (in this
particular aspect) what is called a vector phenomenon.

When we deal with magnitude in relation to the dimensions of space,
our diagram plots magnitude in one direction against magnitude in
another — length against height, for instance, or against breadth ; and
the result is what we call a picture or outhne, or (more correctly)
a "plane projection" of the object. In other words, what we call
Form is a ratio of magnitudes* referred to direction in space.

When, in deahng with magnitude, we refer its variations to
successive intervals of time (or when, as it is said, we equate it with
time), we are then dealing with the phenomenon of growth; and
it is evident that this term growth has wide meanings. For growth
may be positive or negative, a thing may grow larger or smaller,
greater or less; and by extension of the concrete signification of
the word we easily arid legitimately apply it to non-material things,
such as temperature, and say, for instance, that a body "grows"
hot or cold. When in a two-dimensional diagram we represent a
magnitude (for instance length) in relation to time (or "plot" length
against time, as the phrase is), we get that kind of vector diagram
which is known as a "curve of growth." We see that the pheno-
menon which we are studying is a velocity (whose "dimensions" are
space/time, or L/T), and this phenomenon we shall speak of, simply,
as a rate of growth.

In various conventional wa-ys we convert a two-dimensional into

* In Aristotelian logic. Form is a quality. None the less, it is related to qiuirUity't
and we jfind the Schoolmen speaking of it as qualitas circa quantitatem.

CH. Ill] OF CHANGE OF MAGNITUDE 79

a three-dimensional diagram. We do so, for example, when, by
means of the geometrical method of "perspective," we represent
upon a sheet of paper the length, breadth and depth of an object
in three-dimensional space, but we do it better by means of contour-
Hnes or "isopleths." By contour-lines superposed upon a map of
a country, we shew its hills and valleys; and by contour-Unes we
may shew temperature, rainfall, population, language, or any other
*' third dimension" related to the two dimensions of the map. Time
is always impHcit, in so far as each map refers to its own date or
epoch; but Time as a dimension can only be substituted for one of
the three dimensions already there. Thus we may superpose upon
our map the successive outlines of the coast from remote antiquity,
or of any single isotherm or isobar from day to day. And if in hke
manner we superpose on one another, or even set side by side, the
outhnes of a growing organism — for instance of a young leaf and
an old, we have a three-dimensional diagram which is a partial
representation (limited to two dimensions of space) of the organism's
gradual change of form, or course of development; in such a case
our contours may, for the purposes of the embryologist, be separated
by time-intervals of a few hours or days, or, for the palaeontologist,
by interspaces of unnumbered and innumerable years*.

Such a diagram represents in two of its three dimensions form,
and in two (or three) of its dimensions growth, and we see how
intimately the two concepts are correlated or interrelated to one
another. In short it is obvious that the form of an organism is
determined by its rate of growth in various directions ; hence rate
of growth deserves to be studied as a necessary preliminary to the
theoretical study of form, and organic form itself is found,
mathematically speaking, to be di function of time'\.

* Sometimes we find one and the same diagram suffice, whether the time-intervals
be great or small; and we then invoke " Wolff's law" (or Kielmeyer's), and assert
that the life-history of the individual repeats, or recapitulates, the history of the
race. This "recapitulation theory" was alt-important in nineteenth -century
embryology, but was criticised by Adam Sedgwick {Q.J. M.S. xxxvi, p. 38, 1894)
and many later authors; cf. J. Needham, Chemical Embryology, 1931, pp. 1629-1647.

t Our subject is one of Bacon's "Instances of the Course" or studies wherein
we "measure Nature by periods of Time." In Bacpji's Catalogue of Particular
Histories, one of the odd hundred histories or investigations which, he foreshadows
is precisely that which we are engaged on, viz. a "History of the Growth and
Increase of the Body, in the whole and in its parts."

80 THE RATE OF GROWTH [ch.

At the same time, we need only consider this large part of our
subject somewhat briefly. Though it has an essential bearing on
the problems of morphology, it is in greater degree involved with
physiological problems; also, the statistical or numerical aspect of
the question is peculiarly adapted to the mathematical study of
variation and correlation. These important subjects we must not
neglect; but our main purpose will be served if we consider the
characteristics of a rate of growth in a few illustrative cases, and
recognise that this rate of growth is a very important specific
property, with its own characteristic value in this organism or that,
in this or that part of each organism, and in this or that phase of
its existence.

The statement which we have just made that "the form of an
organism is determined by its rate of growth in various directions,"
is one which calls for further explanation and for some measure of
quahfication.

Among organic forms we shall have many an occasion to see that
form may be due in simple cases to the direct action of certain
molecular forces, among which surface-tension plays a leading part.
Now when surface-tension causes (for instance) a minute semifluid
organism to assume a spherical form, or gives to a film of protoplasm
the form of a catenary or of an elastic curve, or when it acts in
various other ways productive of definite contours — just as it does
in the making of a drop, a splash or a jet — this is a process of con-
formation very diiferent from that by which an ordinary plant or
animal grows into its specific form. In both cases change of form
is brought about by the movement of portions of matter, and in
both cases it is ultimately due to the action of molecular forces;
but in the one case the movements of the particles of matter lie for
the most part within molecular range, while in the other we have
to deal with the transference of portions of matter into the system
from without, and from one widely distant part of the organism to
another. It is to this latter class of phenomena that we usually
restrict the term growth; it is in regard to them that we are in a
position to study the rate of action in different directions and at
different times, and to realise that it is on such differences of rate
that form and its modifications essentially and ultimately depend.

Ill] OF RATE OF ACTION 81

The difference between the two classes of phenomena is akin to
the difference between the forces which determine the form of a
raindrop and those which, by the flowing of the waters and the
sculpturing of the solid earth, have brought about the configuration
of a river or a hill ; molecular forces are paramount in the one, and
wolar forces are dominant in the other.

At the same time, it is true that all changes of form, inasmuch
as they necessarily involve changes of actual and relative magnitude,
may in a sense be looked upon as phenomena of growth ; and it is
also true, since the movement of matter must always involve an
element of time*, that in all cases the rate of growth is a phenomenon
to be considered. Even though the molecular forces which play
their part in modifying the form of an organism exert an action
which is, theoretically, all but instantaneous, that action is apt to
be dragged out to an appreciable interval of time by reason of
viscosity or some other form of resistance in the material. From
the physical or physiological point of view the rate of action may be
well worth studying even in such cases as these; for example, a
study of the rate of cell-division in a segmenting egg may teach us
something about the w^ork done, and the various energies concerned.
But in such cases the action is, as a rule, so homogeneous, and the
form finally attained is so definite and so little dependent on the
time taken to effect it, that the specific rate of change, or rate of
growth, does not enter into the morphological problem.

We are deahng with Form in a very concrete way. To Aristotle
it was a metaphysical concept; to us it is a quasi-mechanical effect
on Matter of the operation of chemico-physical forces f. To

* Cf. Aristotle, Phys. VI, 5, 235a, 11, eirel yap awaaa Kiurjffis ev XP^'^V* kt\.; he had
already told us that natural science deals with magnitude, with motion and with
time: ^<ttiu ij irepl (pvaeus eiriar-qtnj irepl fxiyedos Kai klvt^ctlv koL xP^^^^- Hence
omnis velocitas tempore durat became a scholastic aphorism. Bacon emphasised, in
like manner, the fact that "all motion or natural action is performed in time:
some more quickly, some more slowly, but all in periods determined and fixed in
the nature of things. Even those actions which seem to be performed suddenly,
and (as we say) in the twinkling of an eye, are found to admit of degree in
respect of duration" {Nov. Organon, xlvi). That infinitely small motions take
place in infinitely small intervals of time is the concept which lies at the root of the
calculus. But there is another side to the story.

t Cf. N. K. KoltzotF, Physikalisch-chemische Grundlage der Morphologie,
Biol. Centralbl. 1928, pp. 345-369.

82 THE RATE OF GROWTH [ch.

Aristotle its Form was the essence, the archetype, the very "nature"
of a thing, and Matter and Form were an inseparable duahty.
Even now, when we divide our science into Physiology and Mor-
phology, we are harking back to the old Aristotehan antithesis.

To sum up, we may lay down the following general statements.
The form of organisms is a phenomenon to be referred in part to
the direct action of molecular forces, in larger part to a more complex
and slower process, indirectly resulting from chemical, osmotic and
other forces, by which material is introduced into the organism
and transferred from one part of it to another. It is this latter
complex phenomenon which we usually speak of as "growth."

Every growing organism; and every part of such a growing
organism, has its own specific rate of growth, referred to this or
that particular direction ; and it is by the ratio between these rates
in different directions that we must account for the external forms
oiall save certain very minute organisms. This ratio may sometimes
be of a simple kind, as when it results in the mathematically
definable outhne of a shell, or the smooth curve of the margin of a
leaf. It may sometimes be a very constant ratio, in which case the
organism while growing in bulk suffers httle or no perceptible change
in form; but such constancy seldom endures beyond a season, and
when the ratios tend to alter, then we have the phenomenon of
morphological ''development,'' or steady and persistent alteration of
form.

This elementary concept of Form, as determined by varying rates
of Growth, was clearly apprehended by the mathematical mind of
Haller — who had learned his mathematics of the great John
Bemoulh, as the latter in turn had learned his physiology from the
writings of Borelh* It was this very point, the apparently un-
limited extent to which, in the development of the chick, inequalities
of growth could and did produce changes of form and changes of
anatomical structure, that led Haller to surmise that the process
was actually without Hmits, and that all development was but an
unfolding or "evolutio," in which no part came into being which

* "Qua in re Incomparabilis Viri Joh. Alph. Borelli vestigiis insistemus."
Joh. Bernoulli, De motu musculorum, 1694.

Ill] THE DOCTRINE OF PREFORMATION 83

had not essentially existed before*. In short the celebrated doctrine
of "preformation" implied on the one hand a clear recognition of
what growth can do throughout the several stages of development,
by hastening the increase in size of one part, hindering that of
another, changing their relative magnitudes and positions, and so
altering their forms; while on the other hand it betrayed a failure
(inevitable in those days) to recognise the essential difference
between these movements of masses and the molecular processes
which precede and accompany them, and which are characteristic
of another order of magnitude.

The general connection, between growth and form has been
recognised by other writers besides Haller. Such a connection is
imphcit in the "proportional diagrams" by which Diirer and his
brother-artists illustrated the changes in form, or of relative
dimensions, which mark the child's growth to boyhood and to
manhood. The same connection was recognised by the early
embryologists, and appears, as a survival of the doctrine of pre-
formation, in Pander's I study of the development of the chick.
And long afterwards, the embryological aspect of the case was
emphasised by His J, who pointed out that the foldings of the
blastoderm, by which the neural and amniotic folds are brought
into being, were the resultant of unequal rates of growth in what
to begin with was a uniform layer of embryonic tissue. If a sheet
of paper be made to expand here and contract there, as by moisture
or evaporation, the plane surface becomes dimpled, or folded, or
buckled, by the said expansions and contractions; and the dis-
tortions to which the surface of the "germinal disc" is subject are,
as His shewed once and for all, precisely analogous. There are

* Cf. (e.g.) Elem. Physiologiae, ed. 1766, vm, p. 114, "Ducimur autem ad
evolutionem potissimum, quando a perfecto animale retrorsum progredimuis et
incrementonim atque mutationum seriem relegimus. Ita inveniemus perfectum
illud animal fuisse imperfectius, alterius figurae et fabricae, et denique rude et
informe : et tamen idem semper animal sub iis diversis phasibus fuisse, quae absque
ullo saltu perpetuos parvosque per gradus cohaereant."

t Beitrdge zur Entwickelungsgeschichte des Huhnchens im Ei, 1817, p. 40. Roux
ascribes the same views also to Von Baer and to R. H. Lotze {Allgem. Physiologie,
1851, p. 353).

I W. His, Unsere Korperform, und das physiologische Problem ihrer Entstehung,
1874. See also Archiv f. Anatomie, 1894; and cf. C. B. Davenport, Processes con-
cerned in Ontogeny, Bull. Mus. Comp. Anat, xxvn, 1895; also G. Dehnel and
Jan Tur, De Embryonum evolutionis progress u ineqvuli: Kosmos (Lwow), lhi, 1928.

84 THE RATE OF GROWTH [ch.

certain Nostoc-algae in which unequal growth, ceasing towards the
periphery of a disc and increasing here and there within, gives rise
to folds and bucklings curiously Hke those of our own ears: which
indeed owe their shape and characteristic folding to an identical or
analogous cause.

An experimental demonstration comparable to the actual case is
obtained by making an "artificial blastoderm" of Uttle pills or
pellets of dough, which are caused to grow at varying rates by the
addition of varying quantities of yeast. Here, as Roux is careful
to point out,* it is not only the growth of the individual cells, but
the traction exercised on one another through their mutual inter-
connections, which brings about foldings, wrinklings and other
distortions of the structure. But this again, or such as this, had been
in Haller's mind, and formed an essential part of his embry illogical
doctrine. For he has no sooner treated of incrementum, or celeritas
incrementi, than he proceeds to deal with the contributory and
complementary phenomena of expansion, traction (adtractio)^ and
pressure, and the more subtle influences which he denominates vis
derivationis et revulsionis%'. these latter being the secondary and
correlated effects on growth in one part, brought about by such
changes as are produced, for instance in the circulation, by the
growth of another.

We have to do with growth, with exquisitely graded or balanced
growth, and with forces subtly exerted by one growing part upon
another, in so wonderful a piece of work as the development of the
eye : as its primary vesicle expands and then dimples in, as the lens
appears and fits into place, as the secondary vesicle closes over to
form iris and pupil, and in all the rest of the story.

Let us admit that, on the physiological side, Haller's or His's
methods of explanation carry us but a little way; yet even this
little way is something gained. Nevertheless, I can well remember

* Roux, Die Entwickelungsmechanik, 1905, p. 99.

t Op. cit. p. 302, "Magnum hoc naturae instrumentum, etiam in corpore animato
evolvendo potenter operatur, etc." The recurrent laryngeal nerve, drawn down
as its arch of the aorta descends, is a simple instance of anatomical traction. The
vitelline and omphalomesenteric arteries lead, by more complicated constraints
and tractions, to the characteristic loops of the intestinal blood vessels, and of the
intestine itself. Cf. G. Enbom, Lunds Univ. Arsskrift, 1939.

X Ibid. p. 306, "Subtiliora ista, et aliquantum hypothesi mista, tamen magnam
mihi videntur speciem veri habere."

Ill] OF PHYSICS AND EMBRYOLOGY 85

the harsh criticism and even contempt which His's doctrine met
with, not merely on the ground that it was inadequate, but because
such an explanation was deemed wholly inappropriate, and was
utterly disavowed*. Oscar Hertwig, for instance, asserted that, in
embryology, when we find one embryonic stage preceding another,
the existence of the former is, for the embryologist, an all-sufficient
"causal explanation" of the latter. "We consider (he says) that
we are studying and explaining a causal relation when we have
demonstrated that the gastrula arises by invagination of a blasto-
sphere, or the neural canal by the infolding of a cell-plate so as to
constitute a tubef." For Hertwig, then, as Roux remarks, the task
of investigating a physical mechanism in embryology — "der Ziel das
Wirken zu erforschen" — has no existence at all. For Balfour also,
as for Hertwig, the mechanical or physical aspect of organic develop-
ment had little or no attraction. In one notable instance, Balfour
himself adduced a physical, or quasi-physical, explanation of an
organic process, when he referred the various modes of segmentation
of an ovum, complete or partial, equal or unequal and so forth, to
the varying amount or varying distribution of food-yolk associated
with the germinal protoplasm of the egg. But in the main, like all
the other embryologists of his day, Balfour was engrossed in the

* Cf. His, On the Principles of Animal Morphology, Proc. R.S.E. xv,
p. 294, 1888: "My own attempts to introduce some elementary mechanical or
physiolbgical conceptions into embryology have not generally been agreed to by
morphologists. To one it seemed ridiculous to speak of the elasticity of the germinal
layers; another thought that, by such considerations, we 'put the cart before
the horse'; and one more recent author states, that we have better things to do
in embryology than to discuss tensions of germinal layers and similar questions,
since all explanations must of necessity be of a phylogenetic nature. This opposition
to the application of the fundamental principles of science to embryological questions
would scarcely be intelligible had it not a dogmatic background. No other explana-
tion of living forms is allowed than heredity, and any which is founded on another
basis must be rejected.. . .To think that heredity will build organic beings without
mechanical means is a piece of unscientific mysticism." Even the school of
Entwickelungsmechanik showed a certain reluctance, or extreme caution, in speaking
of the physical forces in relation to embryology or physiology. This reluctant
caution is well exemplified by Martin Heidenhain, writing on "Formen und Krafte
in der lebendigen Natur" in Roux's Vortrdge, xxxir, 1923. Speaking of "die
Krafte welche die Entwickelung und den fertigen Zustand der Formen bedingen",
he says: "letztere kann man aber nicht auf dem Felde der Physik suchen, sondern
nur im Umkreis der Lebendigen, obwohl anzunehmen ist, dass diese Krafte spater
einmal ' analogienhaft ' nach dem Vorbilde der Physik beschreibbar sein werden"

t 0. Hertwig, Zeit- und Streitfragen der Biologie, ii, 1897.

86 THE RATE OF GROWTH [ch.

problems of phylogeny, and he expressly defined the aims of com-
parative embryology (as exemphfied in his own textbook) as being
*' twofold: (1) to form a basis for Phylogeny, and (2) to form a basis
for Organogeny, or the origin and evolution of organs*."

It has been the great service of Roux and his fellow-workers of
the school of "Entwickelungsmechanik," and of many other students
to whose work we shall refer, to try, as His tried, to import into
embryology, wherever possible, the simpler concepts of physics, to
introduce along with them the method of experiment, and to refuse
to be bound by the narrow hmitations which such teaching as that
of Hertwig would of necessity impose on the work and the thought
and the whole philosophy of the biologist.

Before we pass from this general discussion to study some of the
particular phenomena of growth, let m.e give an illustration, from
Darwin, of a point of view which is in marked contrast to Haller's
simple but essentially mathematical conception of Form.

There is a curious passage in the Origin of Species "f, where Darwin
is discussing the leading facts of enibryology, and in particular
Von Baer's "law of embryonic resemblance." Here Darwin says:
"We are so much accustomed to see a difference in structure between
the embryo and the adult that we are tempted to look at this
difference as in some necessary manner contingent on growth. But
there is no reason why, for instance, the wing of a bat, or the fin
of a porpoise, should not have been sketched out with all their parts
in proper proportion, as soon as any part became visible." After
pointing out various exceptions, with his habitual care, Darwin
proceeds to lay down two general principles, viz. "that shght
variations generally appear at a not very early period of Hfe," and
secondly, that "at whatever age a variation first appears in the
parent, it tends to reappear at a corresponding age in the offspring."
He then argues that it is with nature as with the fancier, who does
not care what his pigeons look Uke in the embryo so long as the
full-grown bird possesses the desired quahties : and that the process
of selection takes place when the birds or other animals are nearly

* Treatise on Comparative Embryology, i, p. 4, 1881.

t Ist ed. p. 444; 6th ed. p. 390. The student should not fail to consult the
passage in question ; for there is always a risk of misunderstanding or misinterpreta-
tion when one attempts to epitomise Darwin's carefully condensed arguments.

Ill] A PASSAGE IN DARWIN 87

grown up — at least on the part of the breeder, and presumably in
nature as a general rule. The illustration of these principles is set
forth as follows: "Let us take a group of birds, descended from
some ancient form and modified through natural selection for
different habits. Then, from the many successive variations having
supervened iA the several species at a not very early age, and having
been inherited at a corresponding age, the young will still resemble
each other much more closely than do tl^e adults — just as we have
seen with the breeds of the pigeon. . . . Whatever influence long-
continued use or disuse may have had in modifying the limbs or
other parts of any species, this will chiefly or solely have affected
it when nearly mature, when it was compelled to use its full powers
to gain its own living ; and the effects thus produced will have been
transmitted to the offspring at a corresponding nearly mature age.
Thus the young will not be modified, or will be modified only in a
shght degree, through the effects of the increased use or disuse* of
parts." This whole argument is remarkable, in more ways than
we need try to deal with here; but it is especially remarkable that
Darwin should begin by casting doubt upon the broad fact that a
"difference in structure between the embryo and the adult" is
"in some necessary matter contingent on growth"; and that he
should see no reason why compUcated structures of the adult
"should not have been sketched out with all their parts in proper
proportion, as soon as any part became visible." It would seem to
me that even the most elementary attention to form in its relation
to growth would have removed most of Darwin's difficulties in regard
to the particular phenomena which he is considering here. For
these phenomena are phenomena of form, and therefore of relative
magnitude ; and the magnitudes in question are attained by growth,
proceeding with certain specific velocities, and lasting for certain
long periods of time. And it seems obvious accordingly that in any
two related individuals (whether specifically identical or not) the
differences between them must manifest themselves gradually, and
be but Httle apparent in the young. It is for the same simple
reason that animals which are of very different sizes when adult
differ less and less in size (as well as form) as we trace them back-
wards to their early stages.

Though we study the visible effects of varying rates of growth

88 THE RATE OF GROWTH [ch.

throughout wellnigh all the problems of morphology, it is not very
often that we can directly measure the velocities concerned. But
owing to the obvious importance of the phenomenon to the morpho-
logist we must make shift to study it where we can, even though
our illustrative cases may seem sometimes to have little bearing
on the morphological problem*.

In a simple spherical organism, such as the single spherical cell of
Protococcus or of Orbulina, growth is reduced to its simplest terms,
and indeed becomes so simple in its outward manifestations that it
loses interest to the morphologist. The rate of growth is measured
by the rate of change in length of a radius, i.e. F = {R' — R)IT, and
from this we may calculate, as already indicated, the rate in terms
of surface and of volume. The growing body remains of constant
form, by the symmetry of the system; because, that is to say, on
the one hand the pressure exerted by the growing protoplasm is
exerted equally in all directions, after the manner of a hydrostatic
pressure, which indeed it actually is ; while on the other hand the
"skin" or surface layer of the cell is sufficiently homogeneous to exert
an approximately uniform resistance. Under these simple conditions,
then, the rate of growth is uniform in all directions, and does not
affect the form of the organism."

But in a larger or a more complex organism the study of growth,
and of the rate of growth, presents us with a variety of problems,
and the whole phenomenon (apart from its physiological interest)
becomes a factor of great morphological importance. We no longer
find that growth tends to be uniform in all directions, nor have we
any reason to expect it should. The resistances which it meets with
are no longer uniform. In one direction but not in others it will
be opposed by the important resistance of gravity; within the
growing system itself all manner of structural differences come into
play, and set up unequal resistances to growth in one direction or
another. At the same time the actual sources of growth, the
chemical and osmotic forces which lead to the intussusception of
new matter, are not uniformly distributed; one tissue or one organ
may well increase while another does not; a set of bones, their
intervening cartilages and their surrounding muscles, may all be

* "In omni rerum naturalium historia utile est mensuras definiri et numeros,^^
Haller, Elem. Physiol, ii, p. 258, 1760. Cf. Hales, Vegetable Staticks, Introduction.

Ill] ADOLPHE QUETELET 89

capable of very different rates of increment. The changes of form
which result from these differences in rate are especially manifested
during that phase of life when growth itself is rapid: when the
organism^ as we say, is undergoing its development.

When growth in general has slowed down, the differences in rate
between different parts of the organism may still exist, and may be
made manifest by careful observation and measurement, but the
resultant change of form is less apt to strike the eye. Great as are
the differences between the rates of growth in different parts of a
complex organism, the marvel is that the ratios between them are
so nicely balanced as they are, and so capable of keeping the form
of the growing organism all but unchanged for long periods of time,
or of slowly changing it in its own harmonious way. There is the
nicest possible balance of forces and resistances in every part of
the complex body; and when this normal equilibrium is disturbed,
then we get abnormal growth, in the shape of tumours and exostoses,
and other malformations and deformities of every kind.

The rate of growth in man
Man will serve us as well as another organism for our first illus-
trations of rate of growth, nor can we easily find another which we
can better study from birth to the utmost Hmits ofold age. Nor
can we do better than go for our first data concerning him to
Quetelet's Essai de Physique Sociale, an epoch-making book for the
biologist. For it is packed with information, some of it unsurpassed,
in regard to human growth and form; and it stands out as the
first great essay in which social statistics and organic variation are
dealt with from the point of view of the mathematical theory of
probabilities. How on the one hand Quetelet followed Da Vinci,
Luca Pacioli and Dlirer in studying the growth and proportions of
man : and how on the other he simplified and extended the ideas of
James Bernoulli, of d'Alembert, Laplace, Poisson and the rest, is
another and a vastly interesting story*.

* Quetelet, Sur V Homme, ..., ou Essai de Physique Sociale, Bruxelles, 1835:
trans. Edinburgh, 1842; a,\so Instructions populaires sur le calcul des probabilites, 1828;
Lettres. . .sur la theorie des probabilites appliquee aux sciences morales et politiques,
1846 ; and Anthropometrie, 1871 . For an account of his life and writings, see Lottin's
Quetelet, statisticien et sociologue, Louvain, 1912; also J. M. Keynes. Treatise on
Probability, 1921.

90

THE RATE OF GROWTH

[CH.

The meaning of the word "statistics" is curiously changed. For
Shakespeare or for Milton a statist meant (so Dr Johnson says)
"a pohtician, a statesman; one skilled in government." The
eighteenth-century Statistical Account of Scotland was a description
of the State and of its people, its wealth, its. agriculture and its trade.

Stature and weight* of man (from QuMeleVs Belgian data,
Essai, II, pp. 23-43; Anthropometric, p. 346) f

Stature in metres

A

Weight in kgm.

A.

WjU

xlOO

A

Age

Male

Female

% F/M

Male

Female

% F/M

Male

Female

0-50

0-48

96-0

3-20

2-91

90-9

2-56

2-64

1

0-70

0-69

98-6

1000

9-30

93-0

2-92

283

2

0-80

0-78

97-5

1200

11-40

95-0

2-35

2-40

3

0-86

0-85

98-8

13-21

12-45

94-2

2-09

203

4

0-93

0-91

97-6

1507

1418

94-1

1-84

1-88

6

0-99

0-97

98-4

16-70

15-50

92-8

1-89

1-69

6

105

103

98-6

18-04

16-74

92-8

1-56

1-53

1

Ml

MO

98-6

20-16

18-45

91-5

1-48

1-39

8

M7

114

97-3

22-26

19-82

89-0

1-39

1-34

9

1-23

1-20

97-8

24-09

22-44

93-2

1-29

1-30

10

1-28

1-25

97-3

2612

24-24

92-8

1-25

1-24

11

1-33

1-28

96-1

27-85

26-25

94-3

1-18

1-25

12

1-36

1-33

97-6

31-00

30-54

98-5

1-23

1-38

13

1-40

1-39

98-8

35-32

34-65

98-1

1-29

1-29

14

1-49

1-45

97-3

40-50

38-10

94-1

1-21

1-25

15

1-56

1-47

94-6

46-41

41-30

89-0

1-22

1-30

16

1-61

1-52

93-2

53-39

44-44

83-2

1-20

1-32

17

1-67

1-54

92-5

57-40

49-08

85-5

1-23

i-a4

18

1-70

1-56

91-9

61-26

5310

86-.7

1-24

1-40

19

1-71

63-32

1-20

.

20

1-71

1-57

91-8

65-00

54-46

83-8

1-30

1-41

25

1-72

1-58

91-6

68-29

5508

80-7

1-39

1-39

30

172

1-58

91-7

68-90

5514

80-0

1-35

1-39

40

1-71

1-56

90-8

68-81

56-65

82-3

1-38

1-49

50

1-67

1-54

91-8

67-45

58-45

86-7

1-45

1-59

60

1-64

1-52

92-5

65-50

56:73

86-6

1-48

1-61

70

1-62

1-51

93-3

6303

53-72

85-2

1-48

1-58

80

1-61

1-51

93-4

61-22

51-52

84-1

1-46

1-50

This is what Sir Wilham Petty had meant in the seventeenth century
by his Political Arithmetic, and what Quetelet meant in the nine.teenth
by his Physique Sociale. But "statistics" nowadays are counts and
measures of all sorts of things; and statistical science arranges,

* The figures for height and weight given in my first edition were Quetelet's
smoothed or adjusted values. I have gone back to his original data.

t This "almost steady growth," from about seven years old to eleven, means
that the curve of growth is a nearly straight line during this period: a result
already found by Elderton for Glasgow children {Biometrika, x, p. 293, 1914-15),
by Fessard and Laufer in Paris {Nouvelles Tables de Croissance, 1935, p. 13), etc.

Ill]

OF STATISTICAL METHODS

91

explains, and draws deductions from, the resulting series and arrays
of numbers. It deals with simple and measurable effects, due to
complex and often unknown causes ; and when experiment is not at
hand to disentangle these causes, statistical methods may still do
something to elucidate them.

Now as to the growth of man, if the child be some 20 inches, or
say 50 cm., tall at birth, and the man some six feet, or 180 cm.,
high at twenty, we may say that his average rate of growth had

10 13

Time in years

20

Fig. 4. Curve of growth in man. From Quet^let's Belgian data.
The curve H* is proportional to the height cubed.

been (180 - 50)/20 cm., or 6-5 cm. per annum. But we well know
that this is but a rough preliminary statement, and that growth
was surely quick during some and slow during other of those twenty
years; we must learn not only the result of growth but the course
of growth ; we must study it in its continuity. This we do, in the
first instance, by the method of coordinates, plotting magnitude
against time. We measure time along a certain axis (x), and the
magnitude in question along a coordinate axis (y); a succession of
points defines the magnitudes reached at corresponding epochs, and
these points constitute a ''curve of growth'' when we join them
together.

92 THE RATE OF GROWTH [ch.

Our curve of growth, whether for weight or stature, has a definite
form or characteristic curvature: this being a sign that the rate of
growth is not always the same but changes as time goes on. Such
as it is, the curvature alters in an orderly way; so that, apart from
minor and "fortuitous" irregularities, our curves of growth tend to
be smooth curves. And the fact that they are so is an instance of
that "principle of continuity" which is the foundation of all physical
and natural science.

The curve of growth (Fig. 4) for length or stature in man indicates
a rapid increase at the outset, during the quick growth of babyhood ;
a long period of slower but almost steady growth in boyhood; as
a rule a marked quickening in his early teens, when the boy comes
to the "growing age"; and a gradual arrest of growth as he "comes
to his full height" and reaches manhood. If we carried the curve
farther, we should see a very curious thing. We should see that a
man's full stature endures but for a spell; long before fifty* it has
begun to abate, by sixty it is notably lessened, in extreme old age
the old man's frame is shrunken and it is but a memory that "he
once was tall"; the dechne sets in sooner in women than in men,
and "a httle old woman" is a household word. We have seen,
and we see again, that growth may have a negative value, pointing
towards an inevitable end. The phenomenon of negative growth
extends to weight also; it is largely chemical in origin; the meta-
bolism of the body is impaired, and the tissues keep pace no longer
with senile wastage and decay.

We must be very careful, however, how we interpret such a Table
as this; for it records the character of a population, and we are apt
to read in it the life-history of the individual. The two things are
not necessarily the same. That a man grows less as he grows older
all old men know; but it may also be the case, and our Table may
indicate it, that the short men live longer than the tall.

Our curve of growth is, by implication, a "time-energy" diagram f
or diagram of activity. As man grows he is absorbing energy
beyond his daily needs, and accumulating it at a rate depicted in

* Dr Johnson was not far wrong in saying that "life declines from thirty-five";
though the Autocrat of the Breakfast-table declares, like Cicero, that "the furnace
is in full blast for ten years longer".

t J. Joly, The Abundance of Life, 1915 (1890), p. 86.

Ill] OF MAN'S GROWTH AND STATURE 93

our curve ; till the time comes when he accumulates no longer, and
is constrained to draw upon his dwindhng store. But in part, tte
slow decline in stature is a sign of the unequal contest between our
bodily powers and the unchanging force of gravity, which draws us
down when we would fain rise up*; we strive against it all our
days, in every movement of our limbs, in every beat of our hearts.
Gravity makes a difference to a man's height, and no slight one,
between the morning and the evening ; it leaves its mark in sagging
wrinkles, drooping mouth and hanging breasts ; it is the indomitable
force which defeats us in the end, which lays us on our death-bed
and lowers us to the grave f. But the grip in which it holds us is
the title by which we live; were it not for gravity one man might
hurl another by a puff of his breath into the depths of space, beyond
recall for all eternity {.

Side by side with the curve which represents growth in length,
or height or stature, our diagram shews the corresponding curve of
weight. That this curve is of a different shape from the former one
is accounted for in the main (though not wholly) by the fact —
which we have already dealt with — that in similar bodies volume,
and therefore weight, varies as the cubes of the linear dimensions;
and drawing a third curve to represent the cubes of the corresponding
heights, it now resembles the curve of weight pretty closely, but
still they are not quite the same. There is a change of direction,
or "point of inflection," in the curve of weight at one or two years
old, and there are certain other features in our curves which the
scale of the diagram does not make clear; and all these differences
are due to the fact that the child is changing shape as he grows,
that other linear dimensions grow somewhat differently from

* "Lou pes, mestre de tout (Le poids, maitre de tout), m^stre senso vergougno,
Que te tirasso en bas de sa brutalo pougno." J. H. Fabre, Oubreto prouven^alo,
p. 61.

t The continuity of the phenomenon of growth, and the natural passage from
the phase of increase to that of decrease or decay, are admirably discussed by
Enriques, in La Morte, Rivista di Scienza, 1907, and in Wachstum und seine
analytische Darstellung, Biol. Centralbl. June, 1909. Haller [Elementa, vii,
p. 68) recognised decrementum as a phase of growth, not less important (theoretically)
than incrementum ; ''Hristis, sed copiosa, haec est materies."

X Boscovich, Theoria, para. 552, "Homo hominem arreptum a Tellure, et
utcumque exigua impulsum vi vel uno etiam oris flatu impetitum, ab hominum
omnium commercio in infinitum expelleret, nunquam per totam aeternitatem
rediturum."

94

THE RATE OF GROWTH

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Ill] OF CURVES OF GROWTH 95

length or stature, and in short that infant, boy and man are not
similar figures*. The change of form seems sHght and gradual, but
behind it he other and more complex things. Th6 changing ratio
between height and weight imphes changes in the child's metabolism^
in the income and expenditure of the body. The infant stores up
fat, and the active child "runs it off" again; at four years old or
five, bodily metaboUsm and increase of weight are at a minimum;
but a fresh start is made, a new "nutritional period" sets in, and the
small schoolboy grows stout and sJ3rong|.

Our curve of growth shews at successive epochs of time the height
or weight which has been reached by then; it plots changing
magnitude (y) against advancing time (x). It is essentially a
cumulative or summation curve; it sums up or "integrates" all the
successive magnitudes which have been added in all the foregoing
intervals of time. Where the curve is steep it means that growth
was rapid, and when growth ceases the curve becomes a horizontal
line. It follows that, by measuring the slope or steepness of our
curve of growth at successive epochs, we shall obtain a picture of
the successive velocities or growth-rates.

The steepness of a curve is measured by its "gradient J," or we
may roughly estimate it by taking for equal intervals of time
(strictly speaking, for each infinitesimal interval of time) the incre-
ment added during that interval; and this amounts in practice to
taking the differences between the values given for 'the successive
epochs, or ages, which we have begun by studying. Plotting these
successive differences against time, we obtain a curve each point on
which represents a certain rate at a certain time; and while the
former curve shewed a continuous succession of varying magnitudes,
this shews a succession of varying velocities. The mathematician
calls it a curve of first differences ; we may call it a curve of annual
(or other) increments ; but we shall not go wrong if we call it a curve
of the rate (or rates) of growth, or still more simply, a velocity-curve.

* According to Quetelet's data, man's stature is multiplied by 3-4 and his weight
by 20-3, between birth and the age of twenty-one. But the cube of 3-4 is nearly
40; so the weight at birth should be multiplied forty times by the age of
twenty-one, if infant, boy and man were similar figures.

t Cf. T. W. Adams and E. P. Poulton, Heat production in man, Ouy's Hospital
Reports (4), xvn, 1937, and works quoted therein.

I That is, by its trigonometrical tangent, referred to the base-line.

96

THE RATE OF GROWTH

[CH.

We have now obtained two different but closely related curves,
based on the selfsame facts or observations, and illustrating them
in different ways. One is the inverse of the other ; one is the integral
and one the differential of the other; and each makes clear to the
eye phenomena which are imphcit, but are less conspicuous, in the
other. We are using mathematical terms to describe or designate
them; but these "curves of growth" are more comphcated than
the curves with which mathematicians are wont to deal. In our
study of growth we may well hope to find curves simpler than these;

20»

3 5 7 9 II 13 15 17 19 21
Age in years
Fig. 5. Annual increments of growth in man. From Quetelet's Belgian data.

but in the successive annual increments of a boy's growth (as Fig. 5
exhibits them) we are deahng with no one continuous operation
(such as a mathematical formula might define), but with a succession
of events, changing as times and circumstances change.

Our curve of increments, or of first differences, for man's stature
(Fig. 5) is based, perforce, on annual measurements, and growth
alters quickly enough at certain ages to make annual intervals unduly
long; nevertheless our curve shews several important things. It
suffices to shew, for length or stature, that the growth-rate in early
infancy is such as is never afterwards re-attained. From this high
early velocity the rate on the whole falls away, until growth itself

Ill]

OF HUMAN STATURE

97

comes to an end * ; but it does so subject to certain important changes
and interruptions, which are much the same whether we draw them
from Quetelet's Belgian data, or from the British, American and
other statistics of later writers. The curve falls fast and steadily
during the first couple of years of the child's hfe (a). It runs nearly
level during early boyhood, from four or five years old to nine or
ten (6). Then, after a brief but unmistakable period of depression!
during which growth slows down still more (c), the boy enters on

Annual increments of stature and of weight in man
{After Quetelet; see Table, p. 90)

Stature

A

(cm.)

Weight
Male

(kgm.)

Age

Male

Female

Fema

0- 1

20

21

6-8

6-4

1- 2

10

9

2-0

1-9

2- 3

6

7

1-2

11

3- 4

7

6

1-9

1-7

4- 5

6

6

1-6

1-3

5- 6

6

6

1-3

1-2

6- 7

6

7

2-1

1-7

7- 8

6

4

21

1-4

8- 9

6

6

1-8

2-6

9-10

5

5

2-0

1-8

10-11

5

3

1-7

2-0

11-12

3

5

3-2

4-3

12-13

4

6

4-3

4-1

13-14

9

6

5-2

3-5

14-15

7

2

5-9

3-2

15-16

5

3

7-0

2-1

16-17

6

4

40

4-6

17-18

3

2

3-9

40

18-19

1

1

21

1-4

19-20

1-7

—

his teens and begins to "grow out of his clothes"; it is his "growing
age", and comes to its height when he is about thirteen or fourteen
years old (d). The lad goes on growing in stature for some years more,
but the rate begins to fall off (e), and soon does so with great rapidity.
The corresponding curve of increments in weight is not very
different from that for stature, but such differences as there are

* As Haller observed it to do in the chick f "Hoc iterum incrementum miro
ordine distribuitur, ut in principio incubationis maximum est; inde perpetuo
minuatur" {Elementa Pkysiologiae, viii, p. 294). Or as Bichat says, "II y a
surabondance de vie dans I'enfant" {Sur la Vie et la Mort, p. 1).

t This depression, or slowing down before puberty, seems to be a universal
phenomenon, common to all races of men. It is a curious thing that Quetelet's
"adjusted figures" (which I used in my first edition) all but smooth out of
recognition this characteristic feature of his own observations.

98 THE RATE OF GROWTH [ch.

between them are significant enough. There is some tendency for
growth in weight to fall off or fluctuate at four or five years old,
before the small boy goes to school ; but there is, or should be, httle
retardation of weight when growth in height slows down before
he enters on his teens*. The healthy lad puts on weight again
more and more rapidly, for some httle while after gjowth in stature
has slowed down; and normal increase of weight goes on, more
slowly, while the man is "fiUing out," long after growth in stature
has come to an end. But somewhere about thirty he begins losing
weight a httle ; and such subsequent slow changes as men commonly
undergo we need not stop to deal with.

The differences in stature and build between one race and another
are in hke manner a question of growth-rate in the main. Let us
take a single instance, and compare the annual increments of
growth in Chinese and Enghsh boys. The curves are much the
same in form, but differ in amplitude and phase. The Enghsh boy
is growing faster all the while; but the minimal rate and the
maximal rate come later by a year or more than in the Chinese
curve t (Fig. 6).

Quetelet was not the first to study man's growth and stature,
nor was he the first student of social statistics and "demography."
The foundations of modern vital statistics had been laid by Graunt
and Petty in the seventeenth century J; the economists developed
the subject during the eighteenth §, and parts of it were studied

♦ That the annual increments of weight in boys are nearly constant, and the
curve of growth nearly a straight line at this age, especially from about 8 to 11,
has been repeatedly noticed. Cf. Elderton, Glasgow School-children, Biometrika,
X, p. 283, 1914-15; Fessard and others, Croissance des Ecoliers Pariaiens, 1934,
p. 13. But careful measurements of American children, by Katherine Simmons
and T. Wingate Todd, shew steadily increasing increments from four years old
tiU puberty {Growth, n, pp. 93-133, 1938).

t For copious bibliography, see J. Needham, op. cit., also Gaston Backmah, Das
Wachstum der Korperlange des Menschen, K. Sv.Vetensk. Akad. Hdlgr. (3), xiv, 1934.

X Cf. John Graunt's Natural and Political Observations. . .upon the Bills of
Mortality, London, 1662; The Economic Writings of Sir William Petty, ed. by
C. H. Hull, 2 vols,, Cambridge, Mass., 1927. Concerning Graimt and Petty — two
of the original Fellows of the Royal Society — see {int. al.) H. Westergaard, History
of Statistics, 1932, and L. Hogben (and others). Political Arithmetic, 1938.

§ Besides the many works of the economists, cf. J. G. Roederer, Sermo de
pondere et longitudine recens-natorum. Comment. Soe. Beg. Sci. Oottingae, m,
1753; J. F. G. Dietz, De temporum in graviditate et partu aestimatione, Diss.,
Gottingen, 1757.

Ill]

OF HUMAN STATURE

99

7

1 1 1 1 1

1 1 1

1

6

*x N^

-

5

\ y^

\ \
\ \

a 4

^^ '> /

\ \

a

v r

6 ^

Chinese

V

2

\

1

1 f, 1 1 1

1 1 1

1

7

k 9/,o 12/, 3

•6/l7

I7/|8

Age

ig. 6. Annual increments of stature.

From Roberts

(English)

and Appleton's (Chinese) data.

•Date
1760*61 ^62'63'64'65 W67^68*6970'7r72*73'74V5*76 17 77

|6'0

1 2 3 4 5 6 7 8 9 10 n 12 13 14 15 16 17 18

Age in years

Fig. 7. Curve of growth of a French boy of the eighteenth century.

From Scammon, after BufFon.

100

THE RATE OF GROWTH

CH.

eagerly in the early nineteenth, when the exhaustion of the armies
of France and the evils of factory labour in England drew attention
to the stature and physique of man and to the difference between
the healthy and the stunted child*.

A friend of BufPon's, the Count Philibert Gueneau de Montbeillard,
kept careful measurements of his own son; and Buifon pubhshed
these in 1777, in a supplementary volume of the Histoire Naturelle'f.
The child was born in April 1759; it was measured every six months

1
- \

1 1 I

1 1 1

1 ' '

. 1 . .

20

A

—

_ \

-

-

d

-

-

\"

-

-

A

-

10

1

1 1 1

b .

1 1

A

1 1 1 1

1 1

10

20

Fig. 8. Annual increments of stature of the said French boy.

for seventeen years, and the record gives a curve of great interest
and beauty (Fig. 7). There are two ways of studying such a
phenomenon — the statistical method based on large numbers, and
the careful study of the individual case; the curve of growth of this
one French child is to all intents and purposes identical, save that
the boy was throughout a trifle taller, with the mean curve yielded
by a recent study of forty-four thousand Uttle Parisians {.

In young Montbeillard's case the "curve of first differences," or
of the successive annual increments of stature (Fig. 8), is clear and
beautiful. It shews (a) the rapid, but rapidly diminishing, rate of

* Cf. M. Hargenvilliers, Recherches . . . sur . . .le recrutement de Varmie en France,
1817; J. W. Cowell, Measurements of children in Manchester and Stockport,
Factory Reports, i; and works referred to by Quetelet.

t See Richard E. Scammon, The first seriatim study of human growth, Amer.
Journ. of Physical Anthropology, x, pp. 329-336, 1927.

J MM. Variot et Chaumet,- Tables de croissance, dressees. . .d'apr^s les mensura-
tions de 44,000 enfants parisiens, Bull, et M4m. Soc. d'Anthropologie, iii, p. 55,
1906.

Ill] OF GROWTH IN MAN AND WOMAN 101

growth in infancy ; (6) the steady growth in early boyhood ; (c) the
period of retardation which precedes, and (d) the rapid growth which
accompanies, puberty.

Buifon, with his usual wisdom, adds some remarks of his own,
which include two notable discoveries. He had observed that a
man's stature is measurably diminished by fatigue, and the loss soon
made up for in repose; long afterwards Quetelet said, to the same
effect, "le lit est favorable a la croissance, et le matin un homme est
un peu plus grand que le soir." Buff on asked whether growth
varied with the seasons, and Montbeillard's data gave him his reply.
Growth was quicker from April to October than during the rest of
the year: shewing that "la chaleur, qui agit generalement sur le
developpement de tous les etres organisees, influe considerablement
sur I'accroissement du corps humain." Between five years old and
ten, the child grew seven inches during the five summers, but during
the five winters only four; there was a Hke difference again, though
not so great, while the boy was growing quickly in his teens; but
there were no seasonal differences at all from birth to five years old,
when the child was doubtless sheltered from both heat and cold*.

On rate of growth in man and ivoman

That growth follows a different course in boyhood and in girlhood
is a matter of common knowledge; but differences in the curves "of
growth are not very apparent on the scale of our diagrams. They
are better seen in the annual increments, or first differences; and
we may further simplify the comparison by representing the girl's
weight or stature as a percentage of the boy's.

Taking weight to begin with (Fig. 9), the girl's growth-rate is
steady in childhood, from two or three to six or seven years old,

* Growth-rates based on the continuqus study of a single individual are rare;
we depend mostly on average measurements of many individuals grouped according
to their average age. That this is a sound method we take for granted, but we may
lose by it as well as gain. (See above, p. 92.) The chief epochs of growth, the chief
singularities of the curve, will come out much the same in the individual and in the
average curve. But if the individual curves be skew, averaging them will tend to
smooth the skewnesa away; and, more curiously, if they be all more or less diverse,
though all symmetrical, a certain skewness will tend to develop in the composite or
average curve. Cf. Margaret Merrill, The relationship of individual to average
growth, Human Biology, iii. pp. 37-70, 1931.

102

THE RATE OF GROWTH

[CH.

J \ I L

J \ L

2 4 6 8 10 12 14 16 18 19
Age
Fig. 9. Annual increase in weight of Belgian boys and girls.
From Quetelet's data. (Smoothed curves.)

10 15

Age

Fig. 10. Percentage ratio of female weight and stature to male.
From Quetelet's Belgian data.

23

Ill]

OF GROWTH IN MAN AND WOMAN

103

just as is the boy's; but her curve stands on a lower level, for the
little maid is putting on less weight than the boy (b). Later on,
her rate accelerates (c) sooner than does his, but it neyer rises quite
so high {d). After a first maximum at eleven or twelve her rate of
growth slows down a Httle, then rises to a second maximum when

112

110

108

106

104

102

100

98

96

94

92

90

Matle weight / \y /

1. \ L

J I L

1 3 5 7 9 11 13

Fig. i:

Relative weight of American boys and girls.
From Simmons and Todd's data.

she is sixteen or seventeen, after the boy's phase of quickest growth
is over and done. This second spurt of growth, this increase of
vigour and of weight in the girl of seventeen or eighteen, Quetelet's
figures indicate and common observation confirms. Last of all,
while men stop adding to their weight about the age of thirty or
before, this does not happen to women. They increase in weight,
though slowly, till much later on: until there comes a final phase,
in both sexes ahke, when weight and height and strength decline
together.

104

THE RATE OF GROWTH

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Ill

OF GROWTH IN CHILDHOOD

105

Stature and weight of American children (Ohio)
(From Katherine Simmons and T. Wingate Todd's data)

Stature (cm.)

% ratio

Weight (lbs.)
Boys Girls

Age

Boys

Girls

% rati

3 months

61-3

59-3

96-7

14-4

13-1

91-1

1 year

761

. 74-2

97-5

23-9

21-9

91-8

2 „

87-4

86-2

98-6

29-1

27-5

94-7

3 „

96-2

95-5

99-3

33-5

32-5

96-9

4 „

103-9

103-2

99-3

38-4

37-1

96-7

o „

110-9

110-3

99-4

43-2

42-3

98-1

6 „

117-2

117-4

1001

48-5

48-6

100-0

7 „

123-9

123-2

99-4

54-7

54-0

98-8

8 „

130-1

129-3

99-4

62-2

61-5

98-7

9 „

136-0

135-7

99-7

69-5

70-9

102-0

10 „

141-4

140-8

99-6

78-5

77-6

98-8

11 »

146-5

147-8

100-7

86-5

87-0

100-6

12 „

151-1

155-3

102-8

92-7

102-7

110-7

13 „

156-7

159-9

102-0

102-8

114-6

111-4

Mean of observed incremeyits of stature and weight of
American children

Increment of stature (mm.) Increment of weight (lbs.)

''

%

r

/o

Age

Boys

Girls

ratio

Boys

Girls

ratio

3 m

. - 1 yr. 150-4

150-1

99-8

9-32

8-07

93-8

1 yr. - 2 ",

123-9

132-0

106-5

4-97

5-56

1120

2 ,

- 3 ,

88-1

90-0

102-2

4-01

4-18

104-3

3 ,

- 4 ,

73-9

79-1

106-9

4-13

4-46

108-0

4 ,

- 5 ,

69-4

72-2

104-0

4-60

4-55

99-8

5 ,

- 6 ,

67-0

68-0

101-5

4-51

5-08

112-8

6 ,

- 7 ,

64-1

62-6

97-6

5-57

5-40

96-9

7 ,

- 8 ,

61-2

57-8

94-4

6-70

6-65

99-4

8 ,

—9 ,

55-7

60-1

108-0

6-64

7-38

1111

9 ,

-10 ,

54-9

57-7

105-1

7-92

8-12

104-8

10 ,

-11 ,

51-9

61-3

118-2

8-81

9-58

108-7

11 ,

-12 ,

53-2

66-9

125-6

9-54

11-98

1331

12 ,

-13 ,

61-0

55-1

89-0

10-90

10-29

94-7

These differences between the two sexes, which are essentially
phase-differences, cause the ratio between their weights to fluctuate
in a somewhat comphcated way (Figs. 10, 11). At birth the baby
girl's weight is about nine-tenths of the boy's. She gains on him for
a year or two, then falls behind again ; from seven or eight onwards
she gains rapidly, and the girl of twelve or thirteen is very httle
lighter than the boy; indeed in certain American statistics she is
by a good deal the heavier of the two. In their teens the boy gains

106

THE RATE OF GROWTH

[CH.

steadily, and the lad of sixteen is some 15 per cent, heavier than
the lass. The disparity tends to diminish for a while, when the
maid of seventeen has her second spurt of growth ; but it increases
again, though slowly, until at five-and-twenty the young woman is
no more than four-fifths the weight of the man. During middle life
she gains on him, and at sixty the difference stands at some 12

1 1 \ r

J i \ \ \ \ L

2 4 6 8 10 12 14 16 18

Age

Fig. 12. Annual increments of stature, in boys and girls.
From Quetelet's data. (Smoothed curves.)

per cent., not far from the mean for all ages; but the old woman
shrinks and dwindles, and the difference tends to increase again.

The rate of increase of stature, like stature itself, differs notably
in the two sexes, and the differences, as in the case of weight, are
mostly a question of phase (Fig. 12). The httle girl is adding rather
more to her stature than the boy at four years old*, but she grows

* This early spurt of growth in the girl is shewn in Enghsh, French and American
observations, but not in Quetelet's.

Ill

OF GROWTH IN CHILDHOOD

107

slower than he does for a few years thereafter (6). At ten years old
the girl's growth-rate begins to rise (c), a full year before the boy's;
at twelve or thirteen the rate is much ahke for both, but it has
reached its maximum for the girl. The boys' rate goes on rising,
and at fourteen or fifteen they are growing twice as fast as the girls.
So much for the annual increments, as a rough measure of the rates
of growth. In actual stature the baby girl is some 2 or 3 per cent,
below the boy at birth; she makes up the difference, and there is

7 9
Age
Fig. 13. Ratio of female stature to male.

II 13 15

From Simmons and Todd's data.

From R. M. Fleming's data.

good evidence to shew that she is by a very little the taller for a
while, at about five years old or six. At twelve or thirteen she is
very generally the taller of the two, and we call it her "gawky
age" (Fig. 13).

Man and woman differ in length of life, just 3,s they do in weight
and stature. More baby boys are born than girls by nearly 5 peF
cent. The numbers draw towards equality in their teens; after

108 THE RATE OF GROWTH ' [ch.

twenty the women begin to outnumber the men, and at eighty-five
there are twice as many women as men left in the world*.

Men have pondered over the likeness and the unhkeness between
the short lifetimes and the long; and some take it to be fallacious
to measure all alike by the common timepiece of the sun. Life,
they say, has a varying time-scale of its own ; and by this modulus
the sparrow hves as long as the eagle and the day-fly as the manf.
The time-scale of the living has in each case so strange a property
of logarithmic decrement that our days and years are long in
childhood, but an old man's minutes hasten to their end.

On pre-natal and post-natal growth

The rates of growth which we have so far studied are based on
annual increments, or "first differences" between yearly determina-
tions of magnitude. The first increment indicates the mean rate of
growth during the first year of the infant's life, or (on a further
assumption) the mean rate at the mean epoch of six months old;
there is a gap between that epoch and the epoch of birth, of which
we have learned nothing; we do not yet know whether the very
high rate shewn within the first year goes on rising, or tends to fall,
as the date of birth is approached. We are accustomed to inter-
polate freely, and on the whole safely, between known points on
a curve: "si timide que Ton soit, il faut bien que Ton interpole,"
says Henri Poincare; but it is much less safe and seldom justifiable
(at least until we understand the physical principle involved and
its mathematical expression) to "extrapolate" beyond the limits of
our observations.

We must look for more detailed observations, and we may learn
much to begin with from certain old tables of Russow's J, who gives

* Cf. F. E. A. Crew's Presidential Address to Section D of the British Association,
1937.

+ Cf. Gaston Backman, Die organische Zeit, Lunds Universitets Arsskrift, xxxv,
Nr. 7, 1939.

J Quoted in Vierordt's Anatomische . . . Daten und Tabellen, 1906, p. 13. See
also, among many others, Camerer's data, in Pfaundler and Schlossman's Hdb. d.
Kinderheilkunde, i, pp. 49, 424, 1908; Variot, op. cit.; for pre-natal growth, R. E.
Scammon and L. A. Calkins, Growth in the Foetal Period, Minneapolis, 1929. Also,
on this and many other matters, E. Faure-Fremiet, La cinetique du developpement,
Paris, 1925; and, not least, J. Needham, Chemical EinJbryology, 1931.

in]

OF GROWTH IN CHILDHOOD

109

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110

THE RATE OF GROWTH

[CH.

the stature of the infant, month by month, during the first year of
its hfe, as follows :

Mean growth of an infant, in its first twelve-month
(After Russow)

Age (months)

12 3 4

5 6 7 8

9

10 11

12

Length (cm.)
Monthly incre-
ment (cm.)

50 54 58 60 62
— 4 4 2 2

64 65 66 67-5
2 111-5

68
0-5

69 70-5
1 1-5

72
1-5

From these data of Russow's for German children, rough as
indeed they are, from Variot's for little Parisians (Fig. 14), and from

3 10 12

Age in months

Fig. 14. Growth of Parisian children (boys) from birth to twelve months old.
From G. Variot's data; Russow's German data are also shewn, by x x x .

many more, we see that the rate of growth rises steadily and even
rapidly as we pass backwards towards the date of birth. It is never
anything like so great again. It is an impressive demonstration
of the dynamic potentiahty, of the store of energy, in the newborn
child.

But birth itself is but an incident, an inconstant epoch, in the
life and growth of a viviparous animal. The foal and the lamb

Ill] OF PRENATAL GROWTH 111

are born later than a man-child; the puppy and the kitten are born
easier, and in more helpless case than ours; the mouse comes into
the world still earher and more inchoate, so much so that even the
little marsupial is scarcely more embryonic and unformed*. We
must take account, so far as each case permits, of pre-natal or intra-
uterine growth, if we are to study the curve of growth in its entirety.
According to Hisf, the following are the mean lengths from month
to month of the imborn child :

Months 01 23456789 10

(Birth)

Length (mm.) 7-5 40 84 162 275 352 402 443 472 490)

500)

Increment per — 7-5 32-5 44 78 113 77 '50 41 29 18)
month (mm.) 28)

These data hnk on very well to those of Russow, which we have
just considered; and (though His's measurements for the pre-
natal months are more detailed than are those of Russow for the
first year of post-natal hfe) we may draw a continuous curve of
growth (Fig. 15) and of increments of growth (Fig. 16) for the
combined periods. It will be seen at once that there is a "point
of inflection" somewhere about the fifth month of intra-uterine life;
up to that date growth proceeds with a continually increasing
velocity. After that date, though growth is still rapid, its velocity
tends to fall away; the curve, while still ascending, is becoming
an S-shaped curve (Fig. 15). There is a shght break between our
two sets of statistics at the date of birth, an epoch regarding which
we should like to have precise and continuous information. But
we can see that there is undoubtedly a certain shght arrest of growth,
or diminution of the rate of growth, about this epoch ; the sudden
change of nurture has its inevitable effect, but this shght tem-

* It is part of the story, though Ijy no mean? all, that (as Minot says) the larger
the litter the sooner does birth take place. That the day-old foal or fawn can keep
pace with their galloping dams is very remarkable; it is usually explained
teleologically, as a provision of Nature, on which their safety and their survival
depend. But the fact that they come one at a birth has at least something to do
with their comparative maturity.

t Unsere Korperform und das physiologische Problem ihrer Entstehung, Leipzig, 1874.
On growth in weight of the human embryo, see C. M. Jackson, Amer. Journ. Anat.
XVII, p. 118, 1909; also J. Needham, op. cit. pp. 379-383.

112

THE RATE OF GROWTH

[CH.

u

crns

70

-

^^

60

-

^^y^^^'^^

50

/

40

/

-

30

1

Birth

20
10

Lr-r , ,.',.. 1

1

8 10 12 14 16 18 20 22

months

Fig. 15. Curve of growth (in length or stature) of child, before and after birth.
From His and Russow's data.

8 10 12 14

18 20 22
montha

Fig. 16. Mean monthly increments of length or stature of child, in cm.
From His and Russow's data.

Ill] OF GROWTH IN INFANCY 113

porary set-back is immediately followed by a secondary, and equally
transitory, acceleration *.

Mean weight in grams of American infants during ten days
after birth. (From Meredith and Brown)

Weight

Age

f ^ N

(days)

Male

Femal

^t birth

3491

3408

1

3376

3283

2

3294

3207

3

3274

3195

4

3293

3213

5

3326

3246

6

3366

3281

7

3396

3315

8

3421

3341

9

3440

3362

10

3466

3387

Daily

increment

A

Male

Fema

-115

-125

- 82

- 76

- 20

- 12

19

17

33

34

40

35

30

34

25

26

19

21

26

25

The set-back after birth of which we have just spoken is better
shewn by the child's weight than by any linear measurement. During
its first three days the infant loses weight visibly, and it is more than
ten days old before it has made up the weight it lost in those first
three (Fig. 17).

It is worth our while to illustrate on a larger scale His's careful
data for the ten months of pre-natal life (Fig. 18). They give an
S-shaped curve, beautifully regular, and nearly symmetrical on
either side of its point of inflection; and its differential, or curve
of monthly increments, is a bell-shaped curve which indicates with
the utmost simplicity a rise from a minimal to a maximal rate, and
a fall to a minimum again. It has a close family Hkeness to the
well-known "curve of probabihty," of which we shall presently
have much more to say; it is a curve for which we might well
hope to find a simple mathematical expression f-

These two curves, then, look more "mathematical," and less
merely descriptive, than any others we have yet drawn, and much

* See especially, H. V. Meredith and A. W. Brown, Growth in body-weight
during first ten days of postnatal life, Human Biology, xi, pp. 24-77, 1939. Also
{int. al.) T. Brailsford Robertson, Pre- and post-natal growth, etc., Amer. Journ.
Physiol. XXXVII, pp. 1^2, 74-85, 1915.

t The same is not less true of Friedenthal's more elaborate measurements, in his
Physiologie des Menschenwachstums, 1914; cf. Needham, op. cit. p. 1677.

114

THE RATE OF GROWTH

T

CH.

4-100

--100

Fig. 17.

3 10

Age in days
Mean weight of American infants. From Meredith and Brown's data.

500

^^

mm.

^.-"^"^

400

-

y Length

300

-

/

1

200

J

1

100

-

leration

^>^

1 1 1

1

2 4 6 8 10

months
Fig. 18. Curve of a child's pre-natal growth, in length or stature; and corre-
sponding curve of mean monthly increments (mm.). (Smoothed curves.)

Ill] OF GROWTH IN INFANCY 115

the same curves meet us again and again in the growth of other
organisms. The pre-natal growth of the guinea-pig is just the
same*. We have the same essential features, the same S-shaped
curve, in the growth by weight of an ear of maize (Fig. 19), or the
growth in length of the root of a bean (Fig. 20); in both we see
the same slow beginning, the rate rapidly increasing to a maximum,
and the subsequent slowing down or "negative acceleration "f."
One phase passes into another; so far as these curves go, they
exhibit growth as a continuous process, with its beginning, its
middle and its end— a continuity which Sachs recognised some
seventy years ago, and spoke of as the "grand period of growth J."
But these simple curves relate to simple instances, to the infant
sheltered in the womb, or to plant-growth in the sunny season of
the year. They mark a favourable episode, rather than relate the
course of a lifetime. A curve of growth to run all life long is only
simple in the simplest of organisms, and is usually a very complex
affair.

Growth in length of Vallisneria^, and root ofbean\\
and weight of ntaize^

VaUisneria

A

Vicia

Zea

—

-\

Hours Inches

Days

Mm.

Days

Gm.

6 0-3

1-0

6

1

16 1-7

1

2-8

18

4

42 12-6

2

6-5

30

9

54 15-4

3

240

39

17

65 161

4

40-5

46

26

77 16-7

5

57-5

53

42

88 17-1

6

72-0

60

62

7

79-0

74

71

8 790 93 74

It would seem to be a natural rule, that those offspring which
are most highly organised at birth are those which are born largest

* See R. L. Draper, Anat. Record, xviii,"p. 369, 1920; cf. Needham, op. cil.,
p. 1672.

t *^f. R. Chodat et A. Monnier; Sur la courbe de croissance chez les vegetaux.
Bull. Herbier Boissier (2), v, p. 615, 1905.

X Arbeiten a. d. bot. Instit. Wiirzburg, i, p. 569, 1872.

§ A. Bennett, Trans. Linn. Soc. (2), i (Bot.), p. 133, 1880.

II Sachs, I.e.

^ Stefanowska, op. cit. ; G. Backman, Ergebn. d. Physiologie, xxiii, p. 925, 193j

116

THE RATE OF GROWTH

[CH.

20 40 60 80 100

Days
Fig. 19. Growth in weight of maize. From Gustav Backman, after Stefanowska.

5 8

Age in days
Fig. 20. Growth in height of a beanstalk. From Sachs's data.

Ill] OF GROWTH IN INFANCY 117

relatively to their parents' size. But another rule comes in, which
is perhaps less to be expected, that the offspring are born smaller
the larger the species to which they belong. Here we shew, roughly,
the relative weights of the new-born animal and its mother*:

Bear

1:600

Sheep

1:14

Lion

160

Ox

13

Hippopotamus

45

Horse

12

Dog

45-50

Rabbit

40

Cat

25

Mouse ,

10-25

Man

22

Guinea-pig

7

These differences at birth are for the most part made up quickly;
in other words, there are great differences in the rate of growth
during early post-natal life. Two Uourcubs, studied by M. Anthony,
grew as follows:

Male Female

Feb. 23 (born)

—

—

28

2-0 kilos

1-7 kilos

Mar. 8

30

2-6

15

3-8

3-3

22

4-6

4-0

30

5-3

4-6

Apr. 5

61

5-2

12

7-0

6-0

19

8-0

7-0

Thus the lion-cub doubles its weight in the first month, and
wellnigh doubles it again in the second; but the newborn child
takes fully five months to double its weight, and nearly two years
to do so again.

The size finally attained is a resultant of the rate and of the
duration of growth; and one or other of these may be the more
important, in this case or in that. It is on the whole true, as Minot
said, that the rabbit is bigger than the guinea-pig because he grows
faster, but man is bigger than the rabbit because he goes on growing
for a longer time.

A bantam and a barn-door fowl differ in their rate of growth,
which in either case is definite and specific. Bantams have been
bred to match almost every variety of fowl ; and large size or small,
quick growth or slow, is inherited or transmitted as a Mendehan

* Data from Variot, after Anthony,

118 OF THE RATE OF GROWTH [ch.

character in every cross between a bantam and a larger breed.
The bantam is not produced by selecting smaller and smaller
specimens of a larger breed, as an older school might have supposed ;
but always by first crossing with bantam blood, so introducing the
"character" of smallness or retarded growth, and then segregating
the desired types among the dwarfish offspring. In fact. Darwinian
selection plays a small and unimportant part in the process*.

From the whole of the foregoing discussion we see that rate of
growth is a specific phenomenon, deep-seated in the nature of the
organism; wolf and dog, horse and ass, nay man and woman, grow
at different rates under the same circumstances, and pass at different
epochs through like phases of development. Much the same might
be said of mental or intellectual growth ; the girl's mind is more
precocious than the boy's, and its development is sooner arrested
than the man's f.

On variability, and on the curve of frequency or of error

The magnitudes which we are dealing with in .this chapter —
heights and weights and rates of change — are (with few exceptions)
mean values derived from a large number of individual cases. We
deal with what (to borrow a word from atomic physics) we may
call an ensemble; we employ the equaUsing powei^ of averages,
invoke the "law of large numbers J," and claim to obtain results
thereby which are more trustworthy than observation itself §. But
in ascertaining a mean value we must also take account of the
amount of variability, or departure from the mean, among the cases
from which the mean value is derived. This leads on far beyond
our scope, but we must spare it a passing word ; it was this identical
phenomenon, in the case of Man, which suggested to Quetelet the

* Cf. Raymond Pearl, The selection problem, Amer. Naturalist, 1917, p. 82;
R. C. Punnett and P. G. Bailey, Journ. of Genetics, iv, pp. 23-39, 1914.

t Cf. E. Devaux, L'p,llure du developpement dans les deux sexes, Revue gindr.
des Sci. 1926, p. 598.

X S. D. Poisson, following James Bernoulli's Ars Conjectandi (op. posth. 1713),
was the discoverer, or inventor, of the law of large numbers. "Les chos^s
de toute nature sont soumises a une loi universelle qu'on pent appeler la loi des
grands nombres" {Recherches, 1837, pp. 7-12).

§ See p. 137, footnote.

Ill

OF VARIABILITY

119

statistical study of Variation, led Francis Galton to enquire into
the laws of Natural Inheritance, and served Karl Pearson as the
foundation of his science of Biometrics.

When Quetelet tells us that the mean stature of a ten-year-old
boy is 1-275 metres, this is found to imply, not only that the
measurements of all his ten-year-old boys group themselves about
this mean value of 1-275 metres, but that they do so in an orderly
way, many departing httle from that mean value, and fewer and
fewer departing more and more. In fact, when all the measure-
ments are grouped and plotted, so as to shew the number of
instances (y) ^ each gradation of size (x), we obtain a characteristic

-2cr —cr

Fig. 21. The normal curve of frequency, or of error.
a, -a, the "standard deviation".

configuration, mathematically definable, called the curve of frequency,
or oi error (Fig. 21). This is a very remarkable fact. That a "curve
of stature" should agree closely with the "normal curve of error"
amazed Galton, and (as he said) formed the mainstay of his long
and fruitful enquiry into natural inheritance*. The curve is a
thing apart, sui generis. It depicts no course of events, it is no
time or vector diagram. It merely deals with the variabihty, and
variation, of magnitudes; and by magnitudes " we mean anything
which can be counted or measured, a regiment of men, a basket of

* Stature itself, in a homogeneous population, is a good instance of a normal
frequency distribution, save only that the spread or range of variation is unusually
low; for one-half of the population of England differs by no more than an inch
and a half from the average of them all. Variation is said to be greater among the
negroid than among the white races, and it is certainly very great from one race
to another: e.g. from the Dinkas of the White Nile with a mean height of 1-8 m.
to the Congo pygmies averaging 1-35, or say 5 ft. 11 in. and 4 ft. 6 in. respectively.

120 THE RATE OF GROWTH [ch.

niits, the florets of a daisy, the stripes of a zebra, the- nearness of
shots to the bull's eye*. It thereby illustrates one of the most
far-reaching, some say one of the most fundamental, of nature's
laws.

We find the curve of error manifesting itself in the departures
from a mean value, which seems itself to be merely accidental —
as, for instance, the mean height or weight of ten-year-old English
boys; but we find it no less well displayed when a certain definite
or normal number is indicated by the nature of the case. For
instance the Medusae, or jelly-fishes, have a "radiate symmetry"
of eight nodes and internodes. But even so, the number eight is
subject to variation, and the instances of more or less graup them-
selves in a Gaussian curve.

Number of " tentaculocysts'' in Medusae {Ephyra and Aurelia)
[Data from E. T. Browne, QJ.M.S. xxxvii, p. 245, 1895)

5

6

7

8

9

10

11

12

13

14

15

Ephyra (1893) —
„ (1894) 1

4
6

8
34

278
883

22

75

18
61

12
35

14
17

3
3

1

—

Aurelia (1894) —

2

18

296

33

16

18

7

—

—

1

Percentage numbers:
Ephyra —

M

0-5

2-2
30

77-4
79-0

61

6-7

5-0
5-4

3-3
31

3-9
1-4

0-8
0-2

—

—

Aurelia —

0-5

4-7

77-2

8-6

41

2-6

1-8

—

—

—

Mean —

0-7

3-3

77-9

7-1

4-8

3-0

2-4

0-3

—

—

The curve of error is a "bell-shaped curve," a courbe en cloche. It
rises to a maximum, falls away on either side, has neither beginning
nor end. It is (normally) symmetrical, for lack of cause to make it
otherwise; it falls off faster and then slower the farther it departs
from the mean or middle line; it has a "point of inflexion," of
necessity on either side, where it changes its curvature and from
being concave to the middle line spreads out to become convex

* "I know of scarcely anything (says Galton) so apt to impress the imagination
as the wonderful form of cosmic order expressed by the Law of Frequency of
Error. ... It reigns with serenity and in complete self-effacement amidst the
wildest confusion" {Natural Inheritance, p. 62). Observe that Galton calls it the
"law of frequency o/ error," which is indeed its older and proper name. Cf. (int. al.)
P. G. Tait, Trans R.S.E. xxiv, pp. 139-145, 1867.

i

Ill] OF THE CURVE OF ERROR 121

thereto. If we pour a bushel of corn out of a sack, the outhne or
profile of the heap resembles such a curve ; and wellnigh every hill
and mountain in the world is analogous (even though remotely) to
that heap of corn *. Causes beyond our ken have cooperated to place
and allocate each grain or pebble; and we call the result a "random
distribution," and attribute it to fortuity, or chance. Galton
devised a very beautiful experiment, in which a slopmg tray is
beset with pins, and sand or millet-seed poured in at the top.
Every falhng grain has its course deflected again and again; the
final distribution is emphatically a random one, and the curve of
error builds itself up before our eyes.

The curve as defined by Gauss, princeps mathematicorum — who
in turn was building on Laplace f — is at once empirical and
theoretical J ; and Lippmann is said to have remarked to Poincare :
*'Les experimentateurs s'imaginent que c'est un theoreme de
mathematique, et les mathematiciens d'etre un fait experimental ! "
It is theoretical in so far as its equation is based on certam hypo-
thetical considerations: viz. (1) that the arithmetic mean of a number
of variants is their best or likeliest average, an axiom which is
obviously true in simple cases — but not necessarily in all; (2) that
"fortuity" implies the absence of any predominant, decisive or
overwhelming cause, and connotes rather the coexistence and joint
effect of small, undefined but independent causes, many or few:

* If we pour the corn out carefully through a small hole above, the heap becomes
a cone, with sides sloping at an "angle of repose"; and the cone of Fujiyama is an
exquisite illustration of the same thing. But in these two instances one predominant
cause outweighs all the rest, and the distribution is no longer a random one.

t The Gaussian curve of error is really the "second curve of error" of Laplace.
Laplace's first curve of error (which has uses of its own) consists of two exponential
curves, joining in a sharp peak at the median value. Cf. W. J. Luyten, Proc.
Nat. Acad. Sci. xvm, pp. 360-365, 1932.

I The Gaussian equation to the normal frequency distribution or "curve of
error" need not concern us further, but let us state it once for all:

J, _ (xa-x)*

^ V27T

where Xf^ is the abscissa which gives the maximum ordinate, and where the maximum
ordinate, y^ = 1/^/(27t). Thus the log of the ordinate is a quadratic function of the
abscissa ; and a simple property, fundamental to the curve, is that for equally spaced
ordinates (starting anywhere) the square of any ordinate divided by the product of
its neighbours gives a scalar quantity which is constant all along (G.T.B.).

122 THE RATE OF GROWTH [ch.

producing their several variations, deviations or errors; and potent
in their combinations, permutations and interferences*.

We begin to see why bodily dimensions lend, or submit, them-
selves to this masterful law. Stature is no single, simple thing;
it is compounded of bones, cartilages and other elements, variable
each in its own way, some lengthening as others shorten, each
playing its little part, hke a single pin in Galton's toy, towards a
''fortuitous" resultant. "The beautiful regularity in the statures
of a population (says Galton) whenever they are statistically
marshalled in the order of their heights, is due to the number of
variable and quasi-independent elements of which stature is the
sum." In a bagful of pennies fresh from the Mint each coin is
made by the single stroke of an identical die, and no ordinary
weights and measures suffice to differentiate them; but in a bagful
of old-fashioned hand-made nails a slow succession of repeated
operations has drawn the rod and cut the lengths and hammered
out head, shaft and point of every single nail — and a curve of
error depicts the differences between them.

The law of error was formulated by Gauss for the sake of the
astronomers, who aimed at the highest possible accuracy, and
strove so to interpret their observations as to eliminate or minimise
their inevitable personal and instrumental errors. It had its
roots also in the luck of the gaming-table, and in the discovery
by eighteenth-century mathematicians that "chance might be
defined in terms of mathematical precision, or mathematical 'law'."
It was Quetelet who, beginning as astronomer and meteorologist,
applied the "law of frequency of error" for the first time to
biological statistics, with which in name and origin it had nothing
whatsoever to do.

The intrinsic significance of the theory of probabihties and the
law of error is hard to understand. It is sometimes said that to
forecast the future is the main purpose of statistical study, and
expectation, or expectancy, is a common theme. But all the theory

* "The curve of error would seem to carry the great lesson that the ultimate
differences between individuals are* simple and few ; that they depend on collisions
and arrangements, on permutations and combinations, on groupings and inter-
ferences, of elementary qualities which are limited iyi variety and finite in extent''
(J. M. Keynes). A connection between this law and Mendelian inheritance is
discussed by John Brownlee, P.R.S.E. xxxi, p. 251, 1910.

Ill] bF THE LAW OF ERROR 123

in the vorld enables us to foretell no single unknown thing, not even
the turn of a card or the fall of a die. The theory of probabiUties
is a development of the theory of combinations, and only deals with
what occurs, or has occurred, in the long run, among large numbers
and many permutations thereof. Large numbers simplify many
things; a million men are easier to understand than one man out
of a million. As David Hume* said: "What depends on a few
persons is in a great measure to be ascribed to chance, or to secret
and unknown causes; what arises from a great many may often
be accounted for by determinate and known causes." Physics is,
or has become, a comparatively simple science, just because its laws
are based on the statistical averages of innumerable molecular or
primordial elements. In that invisible world we are sometimes told
that "chance" reigns, and "uncertainty" is the rule; but such
phrases as mere chance, or at random, have no meaning at all except
with reference to the knowledge of the observer, and a thing is a
"pure matter of chance" when it depends on laws which we do not
know, or are not considering f. Ever since its inception the merits
and significance of the theory of probabiUties have been variously
estimated. Some say it touches the very foundations of know-
ledge} ; and others remind us that "avec les chiffres on pent tout
demontrer." It is beyond doubt, it is a matter of common ex-
perience, that probability plays its part as a guide to reasoning.
It extends, so to speak, the theory of the syllogism, and has been
called the "logic of uncertain inference "§.

In measuring a group of natural objects, our measurements are
uncertain on the one hand and the objects variable on the other;
and our first care is to measure in such a way, and to such a scale,
that our own errors are small compared with the natural variations.
Then, having made our careful measurements of. a group, we want
to know more of the distribution of the several magnitudes, and

* Essay xiv.

t So Leslie Ellis and G. B. Airy, in correspondence with Sir J. D. Forbes; see
his Life, p. 480.

X Cf. Hans Reichenbach, Les fondements logiques du calcul des probabilit^s,
Annales de Vinst. Poincare, vii, pp. 267, 1937.

§ Cf. J. M. Keynes, A Treatise on Probability, 1921; and A. C. Aitken's Statistical
Mathematics, 1939.

124 THE RATE OF GROWTH [ch.

especially to know two important things. We want a mean value,
as a substitute for the true value* if there be such a thing; let us
use the arithmetic mean to begin with. About this mean the ob-
served values are grouped like a target hit by skilful or unskilful
shots ; we want some measure of their inaccuracy, some measure of
their spread, or scatter, or dispersion, and there are more ways than
one of measuring and of representing this. We do it visibly and
graphically every time we draw the curve (or polygon) of frequency ;
but we want a means of description or tabulation, in words or in
numbers. We find it, according to statistical mathematics, in the
so-called index of variability, or standard deviation (o), which merely
means the average deviation from the meanf. But we must take
some precautions in determining this average; for in the nature of
things these deviations err both by excess and defect, they are
partly positive and partly negative, and their mean value is the mean
of the variants themselves. Their squares, however, are all positive,
and the mean of these takes account of the magnitude of each
deviation with no risk of cancelling out the positive and negative
terms: but the "dimension " of this average of the squares is wrong.
The square root of this average of squares restores the correct
dimension, and the result is the useful index of variability, or of
deviation, which is called o-J.

This standard deviation divides the area under the normal curve
nearly into equal halves, and nearly coincides with the point of
inflexion on either side; it is the simplest algebraic measure of
dispersion, as the mean is the simplest arithmetical measure of
position. When we divide this value by the mean, we get a figure

* It is not always obvious what the "errors" are, nor what it is that they depart
or deviate from. We are apt to think of the arithmetic mean, and to leave it
at that. But were we to try to ascertain the ratio of circumference to diameter
by measuring pennies or cartwheels, our "errors" would be found grouped round
a mean value which no simple arithmetic could define.

t a, the standard deviation, was chosen for its convenience in mathematical
calculation and formulation. It has no special biological significance; and a
simpler index, the "inter-quartile distance," has its advantages for the non-
mathematician, as we shall see presently.

X That is to say : Square the deviation-from-the-mean of each class or ordinate
(^); multiply each by the number of instances (or "variates") in ^that class (/);

divide by the total number (N) ; and take the square-root of the whole : a^= ~ "^ *' .

I

Ill] THE COEFFICIENT OF VARIABILITY 125

which is independent of any particular units, and which is called
the coefficient of variability''^ .

Karl Pearson, measuring the amount of variability in the weight
and height of man, found this coefficient to run as follows : In male
new-born infants, for weight 15-6, and for stature 6-5; in male
adults, for weight 10-8, and for stature 3-6. Here the amount of
variability is thrice as great for weight as for stature among grown
men, and about 2 J times as great in infancy f. The same curious
fact is well brought out in some careful measurements of shell-fish,
as follows;

Variability of youdg Clams (Mactra sp.)X

Average size

A

Coefficient of
variability

Age (years)
Number in sample

1 2
41 20

1 2
41 20

Length (cm.)

Height

Thickness

3-2 6-3
2-3 4-7
1-3 2-8

15-3 6-3
140 6-7

9-6 8-3

Weight (gm.)

6-4 59-8

35-4 18-5

The phenomenon is purely mathematical. Weight varies as the product of
length, height and depth, or (as we have so often seen) as the cube of any one
of these dimensions in the case of similar figures. It is then a mathematical,
rather than a biological fact that, for small deviations, the variability of the
whole tends to be equal to the sum of that of the three constituent dimensions.
For if weight, w, varies as height x, breadth y, and depth z, we may write

w = c.xyz.

„„ ,.„ ,. ,. dw dx dy dz

Whence, ditterentiatmg, = h -^ H .

w X y z

We see that among the shell-fish there is much more variability
in the younger than in the older brood. This may be due to

* It is usually multiplied by 100, to make it of a handier amount; and we may
then define this coefficient, C, AS — ajM x 100.

t Cf. Fr. Boas, Growth of Toronto children,- Rep. of U.S. Cornm. of Education,
1896-7, 1898, pp. 1541-1599; Boas and Clark Wissler, Statist cs of growth,
Education Rep. 1904, 1906, pp. 25-132; H. P. Bowditch Rep. Mass. State Board
of Health, 1877; K. Pearson, On the magnitude of certain coefficients of correlation
in man, Proc. R.S. lxvi, 1900; S. Nagai, Korperkonstitution der Japaner, from
Brugsch-Levy, Biologie d. Person, ii, p. 445, 1928; R. M. Fleming, A study of
growth and development. Medical Research Council, Special Report, No. 190, 1933.

J From F. W. Weymouth, California Fish Bulletin, No. 7, 1923.

126 THE RATE OF GROWTH [ch.

inequality of age; for in a population only a few weeks old, a few
days sooner or later in the date of birth would make more diiference
than later on. But a more important matter, to be seen in man-
kind (Fig. 22), is that variability of stature runs j)ari passu, or
nearly so, with the rate of growth, or curve of annual increments
(cf. Fig. 12). The curve of variability descends when the growth-
rate slackens, and rises high when in late boyhood growth is speeded
up. In short, the amount of variability in stature or in weight is
correlated with, or is a function of, the rate of growth in these
magnitudes.

Judging from the evidence at hand, we may say that variabihty
reaches its height in man about the age of thirteen or fourteen,
rather earlier in the girls than in the boys, and rather earher in the
case of stature than of weight. The difference in this respect between
the boys and the girls is now on one side, now on the other. In
infancy variabihty is greater in the girls; the boys shew it the
more at five or six years old; about ten years old the girls have
it again. From twelve to sixteen the boys are much the more
variable, but by seventeen the balance has swung the other way
(Fig. 23).

Coefficient of variability (ojM x 100) in man, at various ages
Age ... o 6 7 8 9 10 11 12 13 14 15 16 17 18

ytature

I

British (Fleming):

Boys

•51

5-4

50

5-3

5-4

5-6

5-7

5-6

5-8

5-8

5-8

5-0

4-3

30

Girls

5-2/

5-2

50

5-5

5-4

5-6

5-8

5-7

5-6

4-7

4-2

3-9

3-7

3-8

American

4-8

4-6

4-4

4-5

4-4

4-6

4-7

4-9

5-5

5-8

5-6

5-5

4-6

3-7

(Bowditch)

Japanese (Nagai):

Boys

40

—

4-3

—

41

—

40

50

50

4-2

3-2

—

—

Girls

—

4-3

—

41

—

4-5

—

4-5

4-6

3-6

31

30

—

—

Mean

—

4-7

—

4-7

—

4-9

—

^ 5-0

5-3

50

4-6

41

— ■

—

Weight

American

11-6

10-3

Ill

9-9

110

1-6

1-8

13-7

3-6

6-8

15-3

13-3

130

10-4

Japanese :

Boys

10-3

121

—

0-8

—

7-0

51

70

13-8

10-9

—

—

Girls

—

10-2

—

11-2

—

21

—

150

5-6

3-4

11-4

11-5

—

—

Mean — 10-3 — 111 — 11-5 — 11-9 14-8 15-7 13-5 11-9 — --

Ill] THE COEFFICIENT OF VARIABILITY 127

61 1 1 1 1 1 r

1 ^

y r^

American.-^ /
-^ /

Japanese^

I I I I I \ L

5 6 7 8 9 10 II 12 13 14 15 16 17 II
Age
Fig. 22. Variability in stature (boys). After Fleming, Bowditch and Nagai.

+ z

[—

+ 1

t

-

B

ntish

/ /
/ /

//

//

^ \
\ \

\ \
\ \
\ \
\ \

/

/

/
/

\

-1

/

1

Japanese

1

1

3

18

. 10 15

Age in years
Fig. 23. Coefficient of variability in stature: excess or defect of this coefficient
in the boy over the girl. Data from R. M. Fleming, and from Nagai.

128 THE RATE OF GROWTH [ch.

The amount of variability is bound to differ from one race or
nationality to another, and we find big differences between the
Americans and the Japanese, both in magnitude and phase (Fig. 22).

If we take not merely the variability of stature or -weight at a
given age, but the variability of the yearly increments, we find
that this latter variabihty tends to increase steadily, and more and
more rapidly, within the ages for which we have information; and
this phenomenon is, in the main, easy of explanation. For a great
part of the difference between one individual and another in regard
to rate of growth is a mere difference of phase — a difference in the
epochs of acceleration and retardation, and finally a difference as to
the epoch when growth comes to an end ; it follows that variabihty
will be more and more marked as we approach and reach the period
when some individuals still continue, and others have already ceased,
to grow. In the following epitomised table, I have taken Boas's
determinations * of the standard deviation (ct), converted them into
the corresponding coefficients of variabihty {olM x 100), and then
smoothed the resulting numbers:

Coefficients of variability in annual increments of stature

Age ...

7

8

9

10

11

12

13

14

15

Boys
Girls

17-3
171

15-8
17-8

18-6
19-2

191
22-7

21-0
25-9

24-7
29-3

29-0
370

36-2
44-8

461

The greater variabihty in the girls is very marked f, and is
explained (in part at least) by the jnore rapid rate at which the girls
run through the several phases of their growth (Fig. 24). To say that
children of a given age vary in the rate at which they are growing
would seem to be a more fundamental statement than that they
vary in the size to which they have grown.

Just as there is a marked difference in phase between the growth-
curves of the two sexes, that is to say a difference in the epochs
when growth is rapid or the reverse, so also, within each sex, will
there be room for similar, but individual, phase-differences. Thus
we may have children of accelerated development, who at a given

* Op. cit. p. lo48.

I That women are on the whole more variable than men was argued by Karl
Pearson in one of his earlier essays: The Chances of Death and other Studies, 1897.

in

THE COEFFICIENT OF VARIABILITY

129

epoch after birth are growing rapidly and are already "big for their
age"; and .others, of retarded development, who are comparatively
small and have not reached the period of acceleration which, in
greater or less degree, will come to them in turn. In other words,
there must under such circumstances be a strong positive "coefl&cient
of correlation" between stature and rate of growth, and also between

§ 20

7 8 9 10 II 12 13 14 15

Age in years

Fig. 24. Coefficients of variability, in annual increments of stature.
After Boas.

the rate of growth in one year and the next. But it does not by
any means follow that a child who is precociously big will continue
to grow rapidly, and become a man or woman of exceptional
stature *. On the contrary, when in the case of the precocious or
"accelerated" children growth has begun to slow down, the back-

* Some first attempts at analysis seem to shew that the size of the embryo at
birth, or of the seed at germination, has more, influence than we were wont to
suppose on the ultimate size of plant or animal. See (e.g.) Eric Ashby, Heterosis
and the inheritance of acquired characters, Proc. R.S. (B), No. 833, pp. 431-441,
1937; and papers quoted therein.

130 THE RATE OF GROWTH [ch.

ward ones may still be growing rapidly, and so making up (more
or less completely) on the others. In other words, the period of
high positive correlation between stature and increment will tend
to be followed by one of negative correlation. This interesting and
important point, due to Boas and Wissler*, is confirmed by the
following table :

Correlation of stature and increment in boys and girls
(From Boas and Wissler)

Age 6 7 8 9 10 11 12 13 14 15

Stature (B) 112-7 115-5 123-2 127-4 133-2 136-8 142-7 147-3 155-9 162-2

(G) 111-4 117-7 121-4 127-9 131-8 136-7 144-6 149-7 153-8 157-2

Increment (B) 5-7 5-3 4-9 5-1 5-0 4-7 5-9 7-5 6-2 5-2

(G) 5-9 5-5 5-5 5-9 6-2 7-2 6-5 5-4 3-3 1-7

Correlation (B) 0-25 0-11 0-08 0-25 018 0-18 0-48 0-29 -0-42 -0-44

(G) 0-44 0-14 0-24 0-47 018 -0-18 -0-42 -0-39 -0-63 0-11

A minor but very curious point brought out by the same
investigators is that, if instead of stature we deal with height in
the sitting posture (or, practically speaking, with length of trunk
or back), then the correlations between this height and its annual
increment are throughout negative. In other words, there would
seem to be a general tendency for the long trunks to grow slowly
throughout the whole period under investigation. It is a well-
known anatomical fact that tallness is in the main due not to length
of body but to length of limb.

Since growth in height and growth in weight have each, their own
velocities, and these fluctuate, and even the amount of their
variabihty alters with age, it follows that the correlation between
height and weight must not only also vary but must tend to
fluctuate in a somewhat complicated way. The fact is, this corre-
lation passes through alternate maxima and minima, chief among
which are a maximum at about fourteen years of age and a minimum
about twenty-one. Other intercorrelations, such as those between
height or weight and chest-measurement, shew their periodic
variations in like manner; and it is about the time of puberty

* I.e. p. 42, and other papers there quoted. Cf. also T. B. Robertson, Criteria
of Normality in the Growth of Children, Sydney, 1922.

Ill] THE CURVE OF ERROR 131

that correlation tends to be closest, or a norm to be most nearly
approached*.

The whole subject of variabiHty, both of magnitude and rate of
increment, is highly suggestive and instructive: inasmuch as it
helps further to impress upon us that growth and specific rate of
growth are the main physiological factors, of which specific mag-
nitude, dimensions and form are the concrete and visible resultant.
Nor may we forget for a moment that growth-rate, and growth
itself, are both of them very complex things. The increase of the
active tissues, the building of the skeleton and the laying up
of fat and other stores, all these and more enter into the complex
phenomenon of growth. In the first instance we may treat these
many factors as though they were all one. But the breeder and
the geneticist will soon want to deal with them apart; and the
mathematician will scarce look for a simple expression where
so many factors are involved. But the problems of variability,
though they are intimately related to the general problem of
growth, carry us very soon beyond our hmitations.

The curve of error

To return to the curve of error.

The normal curve is a symmetrical one. Its middle point, or
median ordinate, marks the arithmetic mean of all the measurements ;
it is also the mode, or class to which the largest number of individual
instances belong. Mean, median and mode are three diiferent sorts
of average; but they are one and the same in the normal curve.

It is easy to produce a related curve which is not symmetrical,
and in which mean, median and mode are no longer the same.
The heap of corn will be lop-sided or "skew" if the wind be blowing
while the grain is falling: in other words, if some prevailing cause
disturb the quasi-equihbrium of fortuity ; and there are other ways,
some simple, some more subtle, by^ which asymmetry may be
impressed upon our curve.

The Gaussian curve is only one of many similar bell-shaped curves ;
and the binomial coefficients, the numerical coefficients of (a + by,
yield a curve so Hke it that we may treat them as the same. The

* Cf. Joseph Bergson, Growth -changes in physical correlation, Human Biology,
I, p. 4, 1930.

132 THE RATE OF GROWTH [ch.

Gaussian curve extends, in theory, to infinity at either end; and
this infinite extension, or asymptotism, has its biological significance.
We know that this or that athletic record is lowered, slowly but
continually, as the years go by. This is due in part, doubtless, to
increasing skill and improved technique ; but quite apart from these
the record would slowly fall as more and more races are run, owing
to the indefinite extension of the Gaussian curve*.

On the other hand, while the Gaussian curve extends in theory
to infinity, the fact that variation is always limited and that extreme
v3,riations are infinitely rare is one of the chief lessons of the law
of frequency. If, in a population of 100,000 men, 170 cm. be the
mean height and 6 cm. the standard deviation, only 11 per cenl^.,
or say 130 men, will exceed 188 cm., only 10 men will be over
191 cm., and only one over 193 cm., or 13 J per cent, above the
average. The chance is negligible of a single one being found over
210 cm., or 7 ft. high, or 24 per cent, above the average.

Yet, widely as the law holds good, it is hardly safe to count it
as a universal law. Old Parr at 150 years old, or the giant Chang
at more than eight feet high, are not so much extreme instances of
a law of probabihty, as exceptional cases due to some peculiar cause
or influence coming inf. In a somewhat analogous way, one or two
species in a group grow far beyond the average size ; the Atlas moth,
the Gohath beetle, the ostrich and the elephant, are far-off outhers
from the groups to which they belong. A reason is not easy to. find.
It looks as though variations came at last to be in proportion to the
size attained, and so to go on by compound interest or geometrical
progression. There may be nothing surprising in this ; nevertheless,
it is in contradistinction to that summation of small fortuitous
differences which lies at the root of the law of error. If size vary in
proportion to the magnitude of the variant individuals, not only

* This is true up to a certain extent, but would become a mathematical fiction
later on. There will be physical limitations (as there are in quantum mechanics)
both to record-breaking, and to the measurement of minute extensions of the
record.

t We may indeed treat old Parr's case on the ordinary lines of actuarial
probability, but it is "without much actuarial importance." The chance of his
record being broken by a modern centenarian is reckoned at (5)^°, by Major
Greenwood and J. C. Irwin, writing on Senility, in Human Biology, xi, pp. 1-23,
1939.

Ill]

THE CURVE OF ERROR

133

will the frequency curve be obviously skew, but the geometric mean^
not the arithmetic, becomes the most probable value*. Now the
logarithm of the geometric mean of a series of numbers is the
arithmetic mean of their logarithms; and it follows that in such
cases the logarithms of the variants, and not the variants them-
selves, will tend to obey the Gaussian law and follow the normal
curve of frequency f.

The Gaussian curve, and the standard deviation associated with
it, were (as we have seen) invented by a mathematician for the use

21

33

36

24 27 30

Length in mm.
Fig. 25 A. Curve of frequency of a population of minnows.

39

of an astronomer, and their use in biology has its difficulties and
disadvantages. We may do much in a simpler way. Choosing a
random example, I take a catch of minnows, measured in 3 mm.
groups, as follows (Fig. 25 A):

Size (mm.) 13-15 16^18 19-21 22-24 25-27 28^30 31-33 34-36 37-39
Number 1 22 52 67 114 257 177 41 2

* See especially J. C. Kapteyn, Skew frequency curves in biology and statistics,
Rec. des Trav. Botan. N4erland., Groningen, xin, pp. 105-158, 1916. Also Axel
M. Hemmingsen, Statistical analysis of the differences in body-size of related species,
Danske Vidensk. Selsk. Medd. xcvm, pp. 125-160, 1934.

t This often holds good. Wealth breeds wealth, hence the distribution of
wealth follows a skew curve; but logarithmically this curve becomes a normal
one. Weber's law, in physiology, is a well-known instance; on the thresholds
of sensations, effects are produced proportional to the magnitudes of those
thresholds, and the logs of the thresholds, and not the thresholds themselves,
are normaUy distributed.

134

THE RATE OF GROWTH

[CH.

Let us sum the same figures up, so as to show the whole number
above or below the respective sizes.

Size (mm.)

15

18

21

24

27

30

33

36

39

Number below

1

23

75

142

256

513

690

731

733

Percentage

—

31

10-2

19-4

34-9

70-0

941

99-6

100

Our first set of figures, the actual measurements, would give us
the '*courbe en cloche," in the formrof an unsymmetrical (or "skew")
Gaussian curve: one, that is to say, with a long sloping talus on

Extreme
Decile

Quartile

Median

Quartile

Decile
Extreme

Fig. 25 B.

18

33

100%
90

75

.50

36 39

21 24 27 30
Length in mm.
'Curve of distribution" of a population of minnows.

one side of the hilj. The other gives us an "S-shaped curve,'' ap-
parently hmited, but really asymptotic at both ends (Fig. 25 B) ; and
this S-shaped curve is so easy to work with that we may at once divide
it into two halves (so finding the ''median" value), or into quarters
and tenths (giving the "quartiles" and "deciles"), or as we please.
In short, after drawing the curVe to a larger scale, we shall find that
we can safely read it to thirds of a miUimetre, and so draw from it
the following somewhat rough but very useful tabular epitome of
our population of minnows, from which the curve can be recon-
structed at aiiy time:

mm.

13

210

25-3

28-6

30-6

32-3

Extreme
First decile
Lower quartile
Median
Upper quartile
Last decile
Extreme

39

Ill]

OF Multimodal curves

135

This S-shaped "summation-curve" is what Francis Galton called a
curve of distribution, and he "Uked it the better the more he used it."
The spread or "scatter" is conveniently and immediately estimated
by the distance between the two quartiles ; and it happens that this
very nearly coincides with the standard deviation of the normal curve.

40

50 60 70

Micrometer-scale units

80

90

100

Fig. 26. A plankton-sample of fish-eggs: North of Scotland, February 1905.

(Only eggs without oil-globule are counted here.)

A. Dab and Flounder. B, Gadus Esmarckii and G. luscus.

C, Cod and Haddock. D, Plaice.

There are biological questions for which we want all the accuracy
which biometric science can give; but there are many others on
which such refinements are thrown away.

Mathematically speaking, we cannot integrate the Gaussian curve,,
save by using an infinite series; but to all intents and purposes we
are doing so, graphically and very easily, in the illustration we have
just shewn. In any case, whatever may be the precise character of
each, we begin to see how our two simplest curves of growth, the
bell-shaped and the S-shaped curve, form a reciprocal pair, the
integral and the differential of one another "^ — hke the distance travelled

* It is of considerable historical interest to know that this practical method of
summation was first used by Edward Wright, in a Table of Latitudes published in
his Certain Errors in Navigation corrected, 1599, as a means of virtually integrating
sec X. (On this, and on Wright's claim to be the inventor of logarithms, see Florian
Cajori, in Napier Memorial Volume, 1915, pp. 94-99.)

136 THE RATE OF GROWTH [ch.

and the velocit}^ of a moving body. If ^ = e"^^ be the ordinate of
the one, z = le-^'^dx is that of the other.

There is one more kind of frequency-curve which we must take
passing note of. We begin by. thinking of our curve, whether
symmetrical or skew, as the outcome of a single homogeneous
group. But if we happen to have two distinct but intermingled
groups to deal with, differing by ever so little in kind, age, place or
circumstance — leaves of both oak and beech, heights of both men
and women — this heterogeneity will tend to manifest itself in two
separate cusps, or modes, on the common curve: which is then
indeed two curves rolled into one, each keeping something of its
own individuality. For example, the floating eggs of the food-fishes
are much alike, but differ appreciably in size. A random gathering,
netted at the surface of the sea, will yield on measurement a multi-
modal curve, each cusp of which is recognisable, more or less
certainly, as belonging to a particular kind of fish (Fig. 26).

A further note upon curves

A statistical "curve", such as Quetelet seems to have been the
first to use*, is a device whose peculiar and varied beauty we are
apt, through famiharity, to disregard. The curve of frequency which
we have been studying depicts (as a rule) the distribution of mag-
nitudes in a material system (a population, for instance) at a
certain epoch of time ; it represents a given state, and we may call
/it a diagram of configuration "f. But we oftener use our curves
to compare successive states, or changes of magnitude, as one
configuration gives place to another; and such a curve may be
called a diagram of displacement. An imaginary point moves in
imaginary space, the dimensions of which represent those of the
phenomenon in question, dimensions which we may further define
and measure by a system of "coordinates"; the movements of our
point through its figurative space are thus analogous to, and illus-
trative of, the events which constitute the phenomenon. Time is
often represented, and measured, on one of the coordinate axes, and
our diagram of "displacement" then becomes a diagram of velocity.

* In his Theorie des probabilites, 1846.

t See Clerk Maxwell's article "Diagrams," in the Encyclopaedia Britannica,
9th edition.

Ill] OF STATISTICAL CURVES 137

This simple method (said Kelvin) of shewing to the eye the law of
variation, however complicated, of an independent variable, is one
of the most beautiful results of mathematics*.

We make and use our curves in various ways. We set down on
the coordinate network of our chart the points givQ^i by a series of
observations, and connect them up into a continuous series as we
chart the voyage of a ship from her positions day by day ; we may
"smooth" the line, if we so desire. Sometimes we find our points
so crowded, or otherwise so dispersed and distributed, that a line
can be drawn not from one to another but among them all — a method
first used by Sir John Herschel f, when he studied the orbits of the
double stars. His dehcate observations were affected by errors, at
first sight without rhyme or reason, but a curve drawn where the
points lay thickest embodied the common lesson of them all; any
one pair of observations would have sufiiced, whether better or
worse, for the calculation of an orbit, but Herschel's dot-diagram
obtained "from the whole assemblage of observations taken together,
and regarded as a single set of data, a single result in whose favour
they all conspire." It put us in possession, said Herschel, of
something truer than the observations themselves % ; and Whewell
remarked that it enabled us to obtain laws of Nature not only from
good but from very imperfect observations §. These are some
advantages of the use of "curves," which have made them essential
to research and discovery.

It is often helpful and sometimes necessary to smooth our curves,

* Kelvin, Nature, xxix, p. 440, 1884.

t Mem. Astron. Soc. v, p. 171, 1830; Nautical Almanack, 1835, p. 495; etc.

X Here a certain distinction may be observed. We take the average height of a
regiment, because the men actually vary about a mean. But in estimating the place
of a star, or the height of Mont Blanc, we average results which only differ by
I)ersonal or instrumental error. It is this latter process of averaging which leads,
in Herschel's phrase, to results more trustworthy than observation itself. Laplace
had made a similar remark long before {Oeuvres,yii, Theorie des probabilites) : that
we may ascertain the very small effect of a constant cause, by means of a long series
of observations the errors of which exceed the effect itself. He instances the small
deviation to the eastward which the rotation of the earth imposes on a falling body.
In like manner the mean level of the sea may be determined to the second decimal
of an inch by observations of high and low water taken roughly to the nearest inch,
provided these are faithfxilly carried out at every tide, for say a hundred years.
Cf. my paper on Mean Sea Level, in Scottish Fishery Board's Sci. Report for 1915.

§ Novum Organum Renovatum (3rd ed.), 1858, p. 20.

138 THE RATE OF GROWTH [ch.

whether at free hand or by help of m^athematical rules; it is one
way of getting rid of non-essentials — and to do so has been called
the very key-note of mathematics*. A simple rule, first used by
Gauss, is to replace each point by a mean between it and its two
or more neighbours, and so to take a "floating" or "running
average." In so doing we trade once more on the "principle of
continuity"; and recognise that in a series of observations each
one is related to another, and is part of the contributory evidence
on which our knowledge of all the rest depends. But all the while
we feel that Gaussian smoothing gives us a practical or descriptive
result, rather than a mathematical one.

Some curves are more elegant than others. We may have to rest
content with points in which no. order is apparent, as when we plot
the daily rainfall for a month or two; for this phenomenon is one
whose regularity only becomes apparent over long periods, when
average values lead at last to "statistical uniformity." But the
most irregular of curves may be instructive if it coincide with another
not less irregular : as when the curve of a nation's birth-rate, in its
ups and downs, follows or seems to follow the price of wheat or the
spots upon the sun.

It seldom happens, outside of the exact sciences, that we com-
prehend the mathematical aspect of a phenomenon enough to define
(by formulae and constants) the curve which illustrates it. But,
failing such thorough comprehension, we can at least speak of the
trend of our curves and put into words the character and the course
of the phenomena they indicate. We see how this curve or that
indicates a uniform velocity, a tendency towards acceleration or
retardation, a periodic or non-periodic fluctuation, a start from or an
approach to a limit. When the curve becomes, or approximates to,
a mathematical one, the types are few to which it is Kkely to
belong f. A straight line, a parabola, or hyperbola, an exponential
or a logarithmic curve (like x'=ay^), a sine-curve or sinusoid, damped
or no, suffice for a wide range of phenomena; we merely modify our
scale, and change the names of our coordinates.

* Cf. W. H. Young, The mathematic method and its limitations, Atti del Congresso
dei Matematici, Bologna, 192/8, i, p. 203.

t Hence the engineer usually begins, for his first tentative construction, by
drawing one of the familiar curves, catenary, parabola, arc of a circle, or curve of,
sines.

Ill] OF STATISTICAL CURVES 139

The curves we mostly use, other than the Gaussian curve, are
time-diagrams. Each has a beginning and an end; and one and
the same curve may illustrate the life of a man, the economic history
of a kingdom, the schedule of a train between one st^^tion and
another. What it then shews is a velocity, an acceleration, and
a subsequent negative acceleration or retardation. It depicts a
"mechanism" at work, and helps us to see analogous ' mechanisms
in different fields; for Nature rings her many changes on a few
simple themes. The same expressions serve for different orders of
phenomena. The swing of a pendulum, the flow of a current, the
attraction of a magnet, the shock of a blow, have their analogues in
a fluctuation of trade, a wave of prosperity, a blow to credit, a tide
in the affairs of men.

The same exponential curve may illustrate a rate of cooHng, a loss
of electric charge, the chemical action of a ferment or a catalyst.
The S-shaped population-curve or "logistic curve" of Verhulst (to
which we are soon coming) is the hysteresis-curve by which Ewing
represented self-induction in a magnetic field ; it is akin to the path
of a falhng body under the influence of friction; and Lotka has
drawn a curve of the growing mileage of American railways, and
found it to be a typical logistic curve. A few bars of music plotted
in wave-lengths of the notes might be mistaken for a tidal record.
The periodicity of a wave, the acceleration of gravity, retardation
by friction, the role of inertia, the explosive action of a spark or
an electric contact — these are some of the modes of action or "forms
of mechanism" which recur in Hmited number, but in endless shapes
and circumstances*. The way in which one curve fits many
phenomena is characteristic of mathematics itself, which does not deal
with the specific or individual case, but generalises all the while, and
is fond (as Henri Poincare said) of giving the same name to different
things.

Our curves, as we have said, are mostly time-diagrams, and
represent a change in time from one magnitude to another; they are
diagrams of displacement, in Maxwell's phrase. We may consider
four different cases, not equally simple mathematically, but all

* See an admirable little book by Michael Petrovich, Les micanismes communs
aux phenomenes disparates, Paris, 1921.

140 THE RATE OF GROWTH [ch.

capable of explanation, up to a certain point, without mathe-
matics.

(1) If in our coordinate diagram we have merely to pass from
one isolated jpoint to another, a straight line joining the two points
is the shortest — and the hkeliest way.

(2) To rise and fall alternately, going to and fro from maximum
to minimum, a zig-zag rectilinear path would still be, geometrically,
the shortest way; but it would be sharply discontinuous at every
turn, it would run counter to the "principle of continuity," it is not
likely to be nature's way. A wavy course, with no more change of
curvature than is absolutely necessary, is the path which nature
follows. We call it a simple harmonic motion, and the simplest of

The Sine -curve

The S- shaped curve

The bell-shaped curve
Fig. 27. Simple curves, representing a change from one magnitude to another.

all such wavy curves we call a sine-curve. If there be but one
maximum and one minimum, which our variant alternates between,
the vector pathway may be translated into jpolar coordinates; the
vector does- what the hands of the clock do, and a circle takes the
place of the sine-curve.

(3) To pass from a zero-line to a maximum once for all is a very
different thing ; for now minimum and maximum are both of them
continuous states, and the principle of continuity will cause our
vector-variant to leave the one gradually, and arrive gradually at
the other. The problem is how to go uphill from one level road to
another, with the least possible interruption or discontinuity. The
path follows an S-shaped course; it has an inflection midway; and
the first phase and the last are represented by horizontal asymptotes.
This is an important curve, and a common one. It so far resembles

Ill] OF CURVES IN GENERAL 141

an "elastic curve" (though it is not mathematically identical with it)
that it may be roughly simulated by a watchspring, lying between
two parallel straight lines and touching both of them. It has its
kinetic analogue in the motion of a pendulum, which starts from
rest and comes to rest again, after passing midway through its
maximal velocity. It indicates a balance between production and
waste, between growth and decay: an approach on either side to
a state of rest and equilibrium. It shows the speed of a train
between two stations ; it illustrates the growth of a simple organism,
or even of a population of men. A certain simple and symmetrical
case is called the Verhulst- Pearl curve, or the logistic curve.

(4) Lastly, in order to leave a certain minimum, or zero-hne, and
return to it again, the simplest way will be by a curve asymptotic
to the base-line at both ends — or rather in both directions; it will
be a bell-shaped curve, having a maximum midway, and of necessity
a point of inflection on either side ; it is akin to, and under certain
precise conditions it becomes, the curve of error or Gaussian curve.

Besides the ordinary curve of growth, which is a summation-
curve, and the curve of growth-rates, which is its derivative, there
are yet others which we may employ. One of these was introduced
by Minot*, from 'a feeling that the rate of growth, or the amount
of increment, ought in some way to be equated with the growing
structure. Minot's method is to deal, not with the actual increments
added in successive periods, but with these successive increments
represented as percentages of the amount already reached. For
instance, taking Quetelet's values for the height (in centimetres) of
a male infant, we have as follows:

Years

1

2

3

4

era.

500

69-8

791

86-4

92-7

But Minot would state the percentage-growth in each of these
four annual periods at 39-6, 13-3, 9-2 and 7-3 per cent, respectively:

Years 1 2 3 4

Height (cm.) 50-0 69-8 79-1 86-4 92-7

Increments (cm.) — 19-8 9-3 7-3 6-3

(per cent.) ~ 39-6 13-3 9-2 7-3

* C. S. Minot, On certain phenomena of growing old, Proc. Amer. Assoc, xxxix,
1890, 21 pp.; Senescence and rejuvenation, Journ. Physiol, xii, pp. 97-153,
1891; etc. Criticised by S. Brody and J. Needham, ojp, cit. pp. 401 seq.

142 THE RATE OF GROWTH [ch.

Now, in our first curve of growth we plotted length against time,
a very simple thing to do. When we differentiate L with respect to T,
we have dL/dT, which is rate or velocity, again a very simple thing ;
and from this, by a second differentiation, we obtain, if necessary,
d^L/dT^, that is to say, the acceleration.

But when you take percentages of y, you are determining dyjy,
and w^hen you plot this against dx, you have

^, or -^, or -.^.
dx ' y.dx' y' dx'

That is to say, you are multiplying the thing whose variations
you are studying by another quantity which is itself continually
varying; and are dealing with something more complex than the
original factors*. Minot's method deals with a perfectly legitimate
function of x and y, and is tantamount to plotting log y against x,
that is to say, the logarithm of the increment against the time.
This would be all to the good if it led to some simple result, a straight
line for instance ; but it is seldom if ever, as it seems to me, that it
does anything of the kind. It has also been pointed out as a grave
fault in his method that, whereas growth is a continuous process,
Minot chooses an arbitrary time-interval as his basis of comparison,
and uses the same interval in all stages of development. There is
little use in comparing the percentage increase fer week of a week-
old chick, with that of the same bird at six months old or at six
years.

The growth of a population

After dealing with Man's growth and stature, Quetelet turned to
the analogous problem of the growth of a populatfon — all the more
analogous in our eyes since we know man himself to be a "statistical
unit," an assemblage of organs, a population of cells. He had read

* Schmalhausen, among others, uses the same measure of rate of growth, in the
form

log F-logF dvl^
^~ k{t-t) ~^ dt v'

Arch. f. Entw. Mech. cxm, pp. 462-519, 1928.

m] MALTHUS ON POPULATION 143

Malthus's Essay on Population* in a French translation, and was
impressed like all the world by the importance of the theme. He
saw that poverty and misery ensue when a population outgrows its
means of support, and believed that multiphcation is checked both
. by lack of food and fear of poverty. He knew that there were,
and must be, obstacles of one kind or another to the unrestricted
increase of a population; and he knew the more subtle fact that
a population, after growing to a certain height, oscillates about an
unstable level of equilibrium f.

Malthus had said that a population grows by geometrical pro-
gression (as 1, 2, 4, 8) while its means of subsistence tend rather to
grow by arithmetical (as 1, 2, 3, 4) — that one adds up while the
other multiphesj. A geometrical progression is a natural and a

* T. R. Malthus, An Essay on the Principle of Population, as it affects the Future
Improvement of Society, etc., 1798 (6th ed. 1826; transl. by P. and G. Prevost,
Geneva, 1830, 1845). Among the books to which Malthus was most indebted was
A Dissertation on the Numbers of Mankind in ancient and modern Times, published
anonymously in Edinburgh in 1753, but known to be by Robert Wallace and read
by him some years before to the Philosophical Society at Edinburgh. In this
remarkable work the writer says (after the manner of Malthus) that mankind
naturally increase by successive doubling, and tend to do so thrice in a hundred
years. He explains, on the other hand, that "mankind do not actually propagate
according to the rule in our tables, or any other constant rule; ^et tables of this
nature are not entirely useless, but may serve to shew, how much the increase of
mankind is prevented by the various causes which confine their number within
such narrow limits." Malthus was also indebted to David Hume's Political
Discourse, Of the Populousness of ancient Nations, 1752, a work criticised by Wallace.
See also McCulloch's notes to Adam Smith's Wealth of Nations, 1828.

■f" That the nearest approach to equilibrium in a population is long- continued
ebb and flow, a mean level and a tide, was known to Herbert Spencer, and was
stated mathematically long afterwards by Vito Volterra. See also Spencer's First
Principles, ch. ^22, sect. 173: "Every species of plant or animal is perpetually
undergoing a rhythmical variation in number — now from abundance of food and
absence of enemies rising above its average, and then by a consequent scarcity
of food and abundance of enemies being depressed below its average.. . .Amid
these oscillations produced by their conflict, lies that average number of the species
at which its expansive tendency ie in equilibrium with surrounding repressive
tendencies." Cf. A. J. Lotka, Analytical note on certain rhythmic relations in
organic systems, Proc. Nat. Acad. Sci. vi, pp. ^10-415, 1920; but cf. also his Elements
of Physical Biology, 1915, p. 90. An analogy, and perhaps a close one, may be found
on the Bourse or money market.

X That a population will soon oiitrun its means of subsistence was a natural
assumption in Malthus's day, and in his own thickly populated land. The danger
may be postponed and the assumption apparently falsified, as by an Argentine
cattle-ranch /or prairie wheat-farm — but only so long as we enjoy world-wide
freedom of import and exchange.

144 THE RATE OF GROWTH [ch.

common thing, and, apart from the free growth of a population or
an organism, we find it in many biological phenomena. An epidemic
dechnes, or tends to decline, at a rate corresponding to a geometrical
progression; the mortahty from zymotic diseases declines in geo-
metrical progression among children from one to ten years old;
and the chances of death increase in geometrical progression after
a certain time of Hfe for us all*.

But in the ascending scale, the story of the horseshoe nails tells
us how formidable a thing successive multipHcation becomes f.
Enghsh law forbids the protracted accumulation of compound
interest; and hkewise Nature deals after her own fashion with the case,
and provides her automatic remedies. A fungus is growing on an
oaktree — it sheds more spores in a night than the tree drops acorns
in a hundred years. A certain bacillus grows up and multiphes by
two in two hoTlrs' time; its descendants, did they all survive, would
number four thousand in a day, as a man's might in three hundred
years. A codfish lays a million eggs and more — all in order that
one pair may survive to take their parents' places in the world.
On the other hand, the humming-birds lay only two eggs, the auks
and guillemots only one; yet the former are multitudinous in their
haunts, and some say that the Arctic auks and auklets outnumber
all other birds in the world. Linnaeus { shewed that an annual
plant would have a miUion offspring in twenty years, if only two
seeds grew up to maturity in a year.

But multiply as they will, these vast populations have their
limits. They reach the end of their tether, the pace slows down, and
at last they increase no more. Their world is fully peopled, whether
it be an island with its swarms of humming-birds, a test-tube with
its myriads of yeast-cells, or a continent with its miUions of mankind.
Growth, whether of a population or an individual, draws to its
natural end; and Quetelet compares it, by a bold metaphor, to the
motion of a body in a resistant medium. A typical population
grows slowly from an asymptotic minimum; it multiphes quickly;

* According to the Law of Gompertz ; cf. John Brownlee, in Proc. R.S.E. xxxi,
pp. 627-634, 1"911.

t Herbert Spencer, A theory of population deduced from the general law of
animal fertility, Westminster Review, April 1852.

X In his essay De Tellure, 1740.

Ill] VERHULST'S LAW 145

it draws slowly to an ill-defined and asymptotic maximum. The
two ends of the population-curve define, in a general way, the
whole curve between; for so beginning and so ending the curve
must pass through a point of inflection, it must be an S-shaped
curve. It is just such a curve as we have seen imder simple
conditions of growth in an individual organism.

This general and all but obvious trend of a population-curve has
been recognised, with more or less precision, by many writers. It
is imphcit in Quetelet's own words, as follows: "Quand une
population pent se developper hbrement et sans obstacles, elle croit
selon une progression geometrique; si le developpement a heu au
miheu d'obstacles de toute espece qui tendent a Farreter, et qui
agissent d'une maniere uniforme, c'est a dire si I'etat sociale ne
change point, la population n'augmente pas d'une maniere indefinie,
mais elle tend de plus en plus a devenir stationnaire*.'' P. F. Verhulst,
a mathematical colleague of Quetelet's, was interested in the same
things, and tried to give a mathematical shape to the same general
conclusions; that is to say, he looked for a "fonction retardatrice"
which should turn the Malthusian curve of geometrical progression
into the S-shaped, or as he called it, the logistic curve, which should
thus constitute the true "law of population," and thereby indicate
(among other things) the hmit above which the population^ was not
likely to grow f .

Verhulst soon saw that he could only solve his problem in a
prehminary and tentative way; ''la hi de la population nous est
inconnue, parcequ'on ignore la nature de la fonction qui sert de
mesure aux obstacles qui s'opposent a la multiphcation indefini de
I'espece humaine." The materials at hand were almost unbeUevably
scanty and poor. The French statistics were taken from documents
"qui ont ete reconnus entierement fictifs"; in England the growth

* Physique Sociale, i, p. 27, 1835. But Quetelet's brief account is somewhat
ambiguous, and he had in mind a body falling through a resistant medium — which
suggests a limiting velocity, or limiting annual increment, rather than a terminal
value. See Sir G. Udny Yule, The growth of population, Journ. R. Statist. Soc.
Lxxxvm, p. 42, 1925.

t P. F. Verhulst, Notice sur la loi que la population suit dans son accroissement.
Correspondence math. etc. public par M. A. Quetelet, x, pp. 113-121, 1838; Rech.
math, sur la loi etc., Nozw. Mem. de VAcad. R. de Bruxelles, xviii, 38 pp., 1845;
deuxieme Mem., ihid. xx, 32 pp., 1847. The term logistic curve had already been
used by Edward Wright; see antea, p. 135. footnote.

146 THE RATE OF GROWTH [ch.

of the population was estimated by the number of births, and the
births by the baptisms in the Church of England, "de maniere que
les enfants des dissidents ne sont point portes sur les registres
officiels." A law of population, or "loi d'affaibhssement" became
a mere matter of conjecture, and the simplest hypothesis seemed to
Verhulst to be, to regard "cet affaibhssement comme proportionnel
a I'accroissement de la population, depuis le moment ou la difficulty
de trouver de bonnes terres a commence a se faire sentir*."

Verhulst was making two assumptions. The first, which is beyond
question, is that the rate of increase cannot be, and indeed is not,
a constant; and the second is that the rate must somehow depend
on (or be some function of) the population for the time being.
A third assumption, again beyond question, is that the simplest
possible function is a linear function. He suggested as the simplest
possible case that, once the rate begins to fall (or once the struggle
for existence sets in), it will fall the more as the population continues
to grow; we shall have a growth-factor and a retardation-factor in
proportion to one another. He was making early use of a simple
differential equation such as Vito Volterra and others now employ
freely in the general study of natural selec.tionf.

The point wher^ a struggle for existence first sets in, and where
ipso facto the rate of increase begins to diminish, is called by Verhulst
the normal level of the population ; he chooses it for the origin of his
curve, which is so defined as to be symmetrical on either side of
this origin. Thus Verhulst's law, and his logistic curve, owe their
form and their precision and all their power to forecast the future
to certain hypothetical assumptions; and the tentative solution
arrived at is one "sous le point de vue mathematiquej."

♦ Op. cit. p. 8.

t Besides many well-known papers by Volterra, see V. A, Kostitzin, Biologie
mathematique, Paris, 1937. Cf. also, for the so-called "Malaria equations," Ronald
Ross, Prevention of Malaria, 2nd ed. 1911, p. 679; Martini, Zur Epidemiologie d.
Malaria, Hamburg, 1921; W. R. Thompson, C.R. clxxiv, p. 1443, 1922, C. N.
Watson, Nature, cxi, p. 88, 1923.

J Verhulst goes on to say that " une longue serie d 'observations, non interrompues
par de grandes catastrophes sociales ou des revolutions du globe, fera probablement
decouvrir la fonction retardatrice dont il vient d'etre fait mention." Verhulst
simplified his problem to the utmost, but it is more complicated today than ever;
he thought it impossible that a country should draw its bread and meat from
overseas: "lors meme qu'une partie considerable de la population pourrait etre

Ill] THE LOGISTIC CURVE 147

The mathematics of the Verhulst-Pearl curve need hardly concern
us; they are fully dealt with in Raymond Pearl's, Lotka's and other
books. Verhulst starts, as Malthus does, with a population growing
in geometrical progression, and so giving a logarithmic curve:

dp

He then assumes, as his "loi d'affaibUssement," a coefficient of
retardation {n) which increases as the population increases:

dp 2

Integrating, p = ^^_^_^_.

If the point of inflection be taken as the origin, k = 0; and again
for < = 00, jt? = — = L. We may write accordingly:

1 + e-^

Malthus had reckoned on a population doubling itself, if unchecked
by want or "accident," every twenty-five years*; but fifty years
after, Verhulst shewed that this "grande vitesse d'accroissement"
was no longer to be found in France or Belgium or other of the
older countries!, but wa^ still being reaUsed in the United States
(Fig. 28). All over Europe, "le rapport de I'exces annuel des nais-
sances sur les deces, a la population qui I'a fourni, va sans cesse en
s'affaiblissant ; de maniere que Faccroissement annuel, dont la valeur
absolue augmente continuellement lorsqu'il y a progression geo-
metrique, parait suivre une progression tout au plus arithmetique."

nourrie de bles etrangers, jamais un gouvernement sage ne consentira a faire
dependre I'existence de milliers de citoyens du bon vouloir des souverains etrangers."
On this and other problems in the growth of a human population, see L. Hogben's
Genetic Problems, etc., 1937, chap. vii. See^also [int. al.) Warren S. Thompson and
P. K. Whelpton, Population Trends in the United States, 1933; F. Lorimer and
F. Osborn, Dynamics of Population, 1934, etc.

* An estimate based, like the rest of Malthus's arithmetic, on very slender
evidence.

t In Quetelet's time the European countries, far from doubling in twenty-five
years, were estimated to do so in from sixty years (Norway) to four hundred years
(France); see M. Haushofer, Lehrbuch der Statiatik, 1882.

148

THE RATE OF GROWTH

[CH.

The "celebrated aphorism" of Malthus was thus, and to this extent,
confirmed*. In the United States, the Malthusian estimate of
unrestricted increase continued to be reahsed for a hundred years
after Malthus wrote; for the 3-93 millions of the U.S. census of
1790 were doubled three times over in the census of 1860, and four
times over in that of 1890. A capital which doubles in twenty-five
years has grown at 2-85 per cent, per annum, compound interest;
the U.S. population did rather more, for it grew at fully 3 per cent,
for fifty of those hundred years f.

1790 1800

1850

1900

1930

Fig. 28. Population of the United States, 1790-1930.

The population of the whole world and of every continent has
increased during modem times, and the increase is large though
the rate is low. The rate of increase has been put at about half-a-
per-cent per annum for the last three hundred years — a shade more
in Europe and a shade less in the rest of the world { :

* Op. cit. 1845, p. 7.

t Verhulst foretold forty millions as the "extreme limit" of the population
of France, and 6^ millions as that of Belgium. The latter estimate he increased
to 8 millions later on. The actual populations of France and Belgium at the
present time are a little more than the ultimate limit which Verhulst foretold.

+ From A. M. Carr-Saunders' World Population, 1936, p. 30.

in] RAYMOND PEARL 149

An estimate of the population of the world
(After W. F. Willcox)

Mean rate of
1650 1750 1800 1850 1900 increase

Europe 100 140 187 266 401 0-52 % per annum

World total 545 728 906 1171 1608 0-49% „

Verhulst was before his time, and his work was neglected and
presently forgotten. Only some twenty years ago, Raymond Pearl
and L. J. Reed of Baltimore, studying the U.S. population as
Verhulst had done, approached the subject in the same way, and
came to an identical result; then, soon afterwards (about 1924),
Raymond Pearl came across Verhulst's papers, and drew attention
to what we now speak of as the Verhulst-Pearl law. Pearl and
Reed saw, as Verhulst had dope, that a "law of population" which
should cover all the ups and downs of human affairs was not to be
found; and yet the general form which such a law must take was
plain to see. There must be a limit to the population of a region,
great or small; and the curve of growth must sooner or later "turn
over," approach the limit, and resolve itself into an S-shaped curve.
The rate of growth (or annual increment) will depend (1) on the
population at the time, and (2) on "the still unutiUsed reserves of
population-support existing" in the available land. Here we have,
to all intents and purposes, the growth-factor and retardation-factor
of Verhulst, and they lead to the same formula, or the same
differential equation, as his*.

A hundred years have passed since Verhulst dealt with the first
U.S. census returns, and found them verifying the Malthusian
expectation of a doubhng every twenty-five years. That "grande
vitesse d'accroissement " continued through five decennia; but it
ceased some seventy years ago, and a retarding influence has been
manifest through all these seventy years (Fig. 29). It is more
recently, only after the census of 1910, that the curve seemed to be

* Raymond Pearl and L. J. Reed, on the Rate of growth of the population of
the U.S. since 1790, and its mathematical representation, Proc. Nat. Acad. Sci.
VI, pp. 275-288, 1920; ibid, vin, pp. 365-368, 1922; Metron, m, 1923. In the
first edition of Pearl's Medical Biometry and Statistics, 1923 (2nd ed. 1930), Verhulst
is not mentioned. See also his Studies in Human Biology, Baltimore, 1924, Natural
History of Population, 1939, and other works.

150

THE RATE OF GROWTH

[CH.

1800 1850 1900 1940

Fig. 29. Decennial increments of the population of the United States.
* The Civil War. * * The "slump".

zuu

— 1 — 1

-1 — r-

1 1 1 1 1

! I^H^^

-

1

'

•

_

y

y

-

/
/

~

•

150

—

/ _

/

_

1

/
/

¥ ^

-

t

/

§

-

. ^

/

1 '^^

/

f

1

-

-

^

-

-

P

-

50

-

-

"""i

1 1

1 1 1 1 1

1 1 1 1 1 1 r 1

1 1 1

1800 1850 1900 1950 2000

Fig. 30. Conjectural population of the United States,
according to the Verhulst- Pearl Law.

Ill] POPULATION OF THE UNITED STATES 151

finding its turning-point, or point of inflection; and only now, since
1940, can we say with full confidence that it has done so.

A hundred years ago the conditions were still relatively' simple,
but they are far from simple now. Immigration was only beginning
to be an important factor; but immigrants made a quarter of the
whole increase of the population of the United States during eighty
of these hundred years*. Wars and financial crises have made their
mark upon the curve; manners and customs, means and standards
of living, have changed prodigiously. But the S-shaped curve
makes its appearance through all of these, and the Verhulst-Pearl
formula meets the case with surprising accuracy.

Population of the United States

I In ten years

Calculated

No. of

A

Total

■\

Increase
by multi-

by logistic

immigrants

increase of

plication

Population

curve

landed

population Percentage

in 25

Year

xlOOO

(Udny Yule)

xlOOO

xlOOO

increase

years

1790

3,929

3,929

—

—

—

—

1800

5,308

5,336

—

1,379

351

—

1810

7,240

7,223

—

1,932

36-4

1820

9,638

9,757

250

2,398

331

208

1830

12,866

13,109

228

3,228

33-5

205

1840

17,069

17,506

538

4.203

32-7

2-02

1850

23,192

23,192

1,427

6,123

35-9

206

1860

31,443

30,418

2,748

8,251

35-6

210

1870

38,558

39,372

2,123

7,115

32-6

1-91

1880

50,156

50,177

2,741

11,598

301

1-83

1890

62.948

62,769

5,249

12,792

25-5

1-80

1900

75,995

76,870

3,694

13,047

20-7

1-71

1910

91,972

91,972

8,201

15,977

210

1-63

1920

105,711

—

6,347

13,739

14-9

1-52

1930

122,975

—

—

17,264

161

1-46

1940

131,669

—

—

8,694

71

1-33

A colony of yeast or of bacteria is a population in its simplest
terms, and Verhulst's law was rediscovered in the growth of a
bacterial colony some years before Raymond Pearl found it in a
population of men, by Colonel M'Kendrick and Dr Kesava Pai, who put
their case very simply indeed f. The bacillus grows by geometrical

* Without counting the children born to those immigrants after landing, and
before the next census return.

t A. G. M'Kendrick and M. K. Pai, The rate of multiplication of micro-organisms :
a mathematical study, Proc. R.S.E. xxxi, pp. 649-655, 1911. (The period of
generation in B. coli, answering to Malthus's twenty-five years for men, was found
to be 22^ minutes.) Cf. also Myer Coplans, Journ. of Pathol, and Bacleriol. xiv,
p. 1, 1910 and H. G. Thornton, Ana. of Applied Biology, 1922, p. 265.

152 THE RATE OF GROWTH [ch.

progression so long as nutriment is enough and to spare; that is to
say, the rate of growth is proportional to the number present:

dt ^

But in a test-tube colony the supply of nourishment is hmited,
and the rate of multiphcation is bound to fall off. If a be the
original concentration of food-stuff, it will have dwindled by time t
to (a — y). The rate of growth will now be

% = by{<^-y),

which means that the rate of increase iS proportional to the number
of organisms present, and to the concentration of the food-supply.
It is Verhulst's case in a nutshell; the differential equation so
indicated leads to an S-shaped curve which further experiment
confirms; and Sach's "grand period of growth" is seen to accom-
phsh itself*.

The growth of yeast is studied in the everyday routine of a
brewery. But the brewer is concerned only with the phase of
unrestricted growth, and the rules of compound interest are all he
needs, to find its rate or test its constancy. A population of 1360
yeast-cells grew to 3,550,000 in 35 hours: it had multiplied -2610
times. Accordingly,

1^8^51^^^^ = 0-098 = log 1-254.

That is to say, the population had increased at the rate of 25-4 per
cent, per hour, during the 35 hours.

The time (^2) required to double the population is easily found :

log 2 0-301 ^ ^^ ,

* The sigmoid curve illustrates a theorem which, obvious as it may seem, is of no
small philosophical importance, to wit, that a body starting from rest must, in order
to attain a certain velocity, pass through all intermediate velocities on its way.
Galileo discusses this theorem, and attributes it to Plato: "Platone avendo per
avventura avuto concetto non potere alcun mobil passare daUa quiete ad alcun

determinate grado di velocita se non col passare per tutti gli altri gradi di

velocita minori, etc."; Discorsi e dimostrazioni, ed. 1638, p. 254.

Ill] A POPULATION OF YEAST 153

The duplication-period thus determined is known to brewers as
the generation-time.

Much care is taken to ensure the maximal growth. If the yeast
sink to the bottom of the vat only its upper layers enjoy unstinted
nutriment; a potent retardation-factor sets in, and the exponential
phase of the growth-curve degenerates into a premature horizontal
asymptote. Moreover, both the yeast and the bacteria differ in
this respect from the typical (or perhaps only simpHfied) case of
man, that they not only begin to suffer want as soon as there comes
to be a deficiency of any one essential constituent of their food*, but
they also produce things which are injurious to their own growth
and in time fatal to their existence. Growth stops long before the
food-supply is exhausted ; for it does so as soon as a certain balance
is reached, depending on the kind or quahty of the yeast, between
the alcohol and the sugar in the cellf.

If we use the compound-interest law at all, we had better think
of Nature's interest as being paid, not once a year nor once an hour
as our elementary treatment of the yeast-population assumed, but
continuously; and then we learn (in elementary algebra) that in
time t, at rate r, a sum P increases to Pe'^\ or P< ^ ^o^^*.

Applying this to the growth of our sample of 1360 yeast cells,
we have

PtIPo = 2610, log 2610 = 3-417, which, multipHed by the modulus
2-303 = 7-868. Dividing by n = 35, the number of hours,

7-868/35 = 0-225 = r.

The rate, that is to say, is 22-5 per cent, per hour, continuous
compound interest. It becomes a well-defined physiological constant,
and we may call it, with V. H. Blackman, an index of efficiency.
Our former result, for interest at hourly intervals, was 25-4 per

* According to Liebig's "law of the minimum."

t T. Carlson, Geschwindigkeit und Grosse'der Hefevermehrung, Biochem. Ztschr.
Lvn, pp. 313-334, 1913; A. Slator, Journ. Chem. Soc. cxix, pp. 128-142, 1906;
Biochem. Journ. vii, p. 198, 1913; 0. W. Richards, Ann. of Botany, xlh, pp. 271-283,
1928; Alf Klem, Hvalradets Skrifter, nr. 7, pp. 55-91, Oslo, 1933; Per Ottestad,
ibid. pp. 30-54. For optimum conditions of temperature, nutriment, pB., etc. see
Oscar W. Richards, Analysis of growth as illustrated by yeast. Cold Spring Harbour
Symposia, ii, pp. 157-166, 1934.

154

THE RATE OF GROWTH

[CH.

cent. ; there is no great difference between such short intervals and
actual continuity, but there is a deal of difference between continuous
payment and payment (say) once a year*. Certain sunflowers
(Helianihus) were found to grow as follows, in thirty-seven days:

Compound interest rate (%)

Weight (gm.)

Continuous

Discontinuous

Giant sunflower
Dwarf sunflower

Seedling

0033
0035

Plant

17-33
14-81

Per day Per wk. Per day Per wk.

170 119 18-5 228%

16-4 114 17-7 214%

When the yeast population is allowed to run its course, it yields
a simple S-shaped curve ; and the curve of first differences derived

rTi*-2*5<rm-2(r

m-o"

m

m-fcr

m+2(r m+2*5(r

(

1
3 1

2

3

4

5 6 7

Fig. 31. The growth of a yeast-population. After Per Ottestad.

from this is, necessarily, a bell-shaped curve, so closely resembling
the Gaussian curve that any difference between them becomes a
deUcate matter. Taking the numbers of the population at equal
intervals of time from asymptotic start to asymptotic finish, we
may treat this series of numbers like any other frequency distribu-
tion. Finding in the usual way the mode and standard deviation,

* Cf. V. H. Blackman, The compound interest law and plant growth, Ann. of
Botany, xxxiii, pp. 353-360, 1919. The first papers on growth by compound
interest in plants were by pupils of Noll in Bonn: e.g. von Kreusler, Wachstum der
Maispflanze, Landw. JB. 1877-79; P. Gressler, Suhstanz-quotienten von Helianthus,
Diss. Bonn, 1907 etc.

Ill] A POPULATION OF FLIES 155

we draw the corresponding Gaussian curve; and the close "fit"
between the observed population-curve and the calculated Gaussian
curve is sufficiently shewn by Mr Per Ottestad's figure (Fig. 31).
This is a very remarkable thing. We began to think of the curve
of error as a function with which time had nothing to do, but here
we have the same curve (or to all intents and purposes the same)
with time for one of its coordinates. We might (I think) add one
more to the names of the curve of error, and call it the curve
of optimum ; it represents on either hand the natural passage from
best to worst, from Ukehest to least likely.

A few flies (Drosophila) in a bottle illustrate the rise and fall of
a population more complex than yeast, as Raymond Pearl has
shewn* The colony dwindles to extinction if food be v/ithheld;
if it be sufficient, the numbers rise in a smooth S-shaped curve;
if it be plentiful and of the best, they end by fluctuating about an
unstable maximum. "The population waves up and down about
an average size," as Raymond Pearl says, as Herbert Spencer had
foreseen t, and as Vito Volterra's differential equations explain.
The growth-rate slackens long before the hunger hne is reached;
crowding affects the birth-rate as well as the death-rate, and a
bottleful of flies produces fewer and fewer offspring per pair the
more flies we put into the bottle {. It is true also of mankind, as
Dr WiUiam Farr was the first to shew, that overcrowding diminishes
the birth-rate and shortens the "expectation of Hfe§ ." It happened
so in the United States, pari passu with the growth of immigration,
incipient congestion acting (or so it seemed) as an obstacle, or a
deterrent, to the large families of former days. Nevertheless, children
still pullulate in the slums. The struggle for existence is no simple
affair, and things happen which no mathematics can foretell.

* Raymond Pearl and S. L. Parker, in Proc. Nat. Acad. Sci. vni, pp. 212-219,
1922; Pearl, Journ. Exper. Zool. Lxm, pp. 57-84, 1932.

t "Wherever antagonistic forces are in^ action, there tends to be alternate
predominance."

X In certain insects an optimum density has been observed; a certain amount
of crowding accelerates, and a greater amount retards, the rate of reproduction.
Cf. D. Stewart Maclagan, Effect of population-density on rate of reproduction,
Proc. R, S. (B), CXI, p. 437, 1932; W. Goetsch, Ueber wachstumhemmende Factoren,
Zool. Jakrb. (Allg. Zool.), xlv, pp. 799-840, 1928.

§ Dr W. Farr, Fifth Report of the Registrar-General, 1843, p. 406 (2nd ed.).

156 THE RATE OF GROWTH [ch.

An analogous S-shaped curve, given by the formula L^ = kg^,
was introduced by Benjamin Gompertz in 1825* ; it is well known to
actuaries, and has been used as a curve of growth by several writers
in preference to the logistic curve. It was devised, and well devised,
to express a "law of human mortality", and to signify the number
surviving at any given age (x), "if the average exhaustions of a
man's power to avoid death were such that at the end of infinitely
small intervals of time he lost equal portions (i.e. equal proportions)
of his remaining power 'to oppose destruction." The principle
involved is very important. Death comes by two roads. One is
by cha^ce or accident, the other by a steady deterioration, or
exhaustion, or growing inability to withstand destruction; and
exhaustion comes (roughly speaking) as by the repeated strokes of
an air-pump, for the life-tables shew mortality increasing in geo-
metrical progression, at least to a first approximation and over
considerable periods of years. Gompertz relied wholly on the
experience of "life-contingencies," but the same deterioration of
bodily energies is plainly visible as growth itself slows down; for
we have seen how growth-rate in infancy is such as is never after-
wards attained, and we may speak of growth-energy and its gradual
loss or decrement, by an easy but significant alteration of phrase.
To deal with the declining growth-rate, as Gompertz did with the
falling expectation of life, and so to measure the remaining energy
available from time to time, would be a greater thing than to record
mere weights and sizes; it raises the problem from mere change of
physical magnitudes to an estimation of the falling or fluctuating
physiological energies of the bodyf. We have seen how in only

* Benjamin Gompertz, On the nature of the function expressive of the law of
human mortality, Phil. Trans, xxxvi, pp. 513-585, 1825. First suggested for use
in growth-problems by Sewall Wright, Journ. Amer. Statist, Soc. xxi, p. 493,
1926. See also C. P. Winsor, The Gompertz curve as a growth curve, Proc. Nat.
Acad. Sci. XVIII, pp. 1-8, 1932; cf. {int. al.) G. R. Da vies. The growth curve,
Journ. Amer. Statist. Soc. xxii, pp. 370-374, 1927; F. W. Weymouth and S. H.
Thompson, Age and growth of the Pacific cockle, Bull. Bureau Fisheries, xlvi,
pp. 63f3-641, 1930-31 ; also Weymouth, McMillen and Rich, in Journ. Exp. Biol.
vni, p. 228, 1931.

t A bold attempt to treat the question from the physiological side, and on
Gompertz's lines, was made only the other day by P. B. Medawar, The growth,
growth-energy and ageing of the chicken's heart, Proc. R.S. (B), cxxix, pp. 332-
355, 1940. Cf. James Gray, The kinetics of growth, Journ. Exp. Biol, vi, pp. 248-
274, 1929.

Ill] THE GOMPERTZ CURVE 157

few and simple cases can a simple curve or single formula be found
to represent the growth-rate of an organism; and how our curves
mostly suggest cycles of growth, each spurt or cycle enduring for
a time, and one following another. Nothing can be more natural
from the physiological point of view than that energy should be
now added and now withheld, whether with the return of the
seasons or at other stages on the eventful journey from childhood
to manhood and old age.

The symmetry, or lack of skewness, in the Verhulst-Pearl logistic
curve is a weak point rather than a strong; the Gompertz curve
is a skew curve, with its point of inflexion not half-way, but about
one- third of the way between the asymptotes. But whether in
this or in the logistic or any other equation of growth, the precise
point of inflexion has no biological significance whatsoever. What
we want, in the first instance, is an S-shaped curve with a variable,
or modifiable, degree of skewness. After all, the same difficulty
arises in all the use we make of the Gaussian curve : which has to
be eked out by a whole family of skew curves, more or less easily
derived from it. We are far from being confined to the Gaussian
curve {sensu stricto) in our studies of biological probabihty, or to the
logistic curve in the study of population.

Yet another equation has been proposed to the S-shaped curve
of growth, by Gaston Backman, a very dihgent student of the
whole subject. The rate of growth is made up, he says, of three
components: a constant velocity, an acceleration varying with the
time, and a retardation which we may suppose to vary with the
square of the time. Acceleration would then tend to prevail in the
earlier part of the curve, and retardation in the latter, as in fact
they do; and the equation to the curve might be written:

log H = ko + k^ log T-k^ log2 T.

The formula is an elastic one, and can be made to fit many an
S-shaped curve; but again it is empirical.

The logistic curve, as defined by Verhulst and by Pearl, has
doubtless an interest of its own for the mathematician, the statistician
and the actuary. But putting aside all its mathematical details and
all arbitrary assumptions, the generalised S-shaped curve is a very
symbol of childhood, maturity and age, of activity which rises to

158 THE RATE OF GROWTH [ch.

fall again, of growth which has its sequel in decay. The growth
of a child or of a nation; the history of a railway*, or the speed
between stations of a train; the spread of an epidemic]", or the
evolutionary survival of a favoured type J — all these things run
their course, in its beginning, its middle and its end, after the fashion
of the S-shaped curve. That curve represents a certain common
pattern among Nature's "mechanisms," and is (as we have said
before), a "mecanisme commun aux phenomenes disparates§."

At the same time — and this is a very interesting part of the story
— the S-shaped curve is no other than what Galton called a curve of
distribution, that is to say a curve of integration or summation-
curve, whose differential is closely akin to the Gaussian curve of
error.

Such, to a first approximation, is our S-shaped population-curve,
and such are the many phenomena which, to a first approximation,
it helps us to compare. But it is only to a first approximation that
we compare the growth of a population with that of an organism,
or for that matter of one organism or one pbpulation with another.
There are immense differences between a simple and a complex
organism, between a primitive and a civihsed population. The
yeast-plant gives a growth-curve which we can analyse; but we
must fain be content with a qualitative description of the growth
of a complex organism in its complex world ||.

There is a simphcity in a colony of protozoa and a complexity in
a warm-blooded animal, a uniformity in a primitive tribe and a
heterogeneity in a modern state or town, which affect all their
economies and interchanges, all the relations between milieu interne
and ex^rne. and all the coefficients in any but the simplest equations of
growth which we can ever attempt to frame. Every growth-problem
becomes at last a specific one, running its own course for its own
reasons. Our curves of growth are all alike — but no two are ever

* Raymond Pearl, Amer. Nat. lxi, pp. 289-318, 1927.

t Ronald Ross, Prevention of Malaria (2nd ed.), 1911, p. 679.

j J. B. S. Haldane, Trans. Camb. Phil. Soc. xxm, pp. 19^1, 1924.

§ Cf. {int. al.) J. R. Miner's Note on birth-rate and density in a logistic population,
Human Biology, iv, p. 119, 1932; and cf. Lotka, ibid, in, p. 458, 1931.

II Cf. {int. al.) C. E. Briggs, Attempts to analyse growth- curves, Proc. E.S. (B),
en, pp. 280-285, 1928.

Ill] IN VARIOUS ORGANISMS 159

the same. Growth keeps caUing our attention to its own com-
plexity. We see it in the rates of growth which change with age
or season, which vary from one hmb to another,; in the influence
of peace and plenty, of war and famine ; not least in those composite
populations whose own parts aid or hamper one another, in any
form or aspect of the struggle for existence. So we come to the
differential equations, easy to frame, more difficult to solve, easy in
their first steps, hard and very powerful later on, by which Lotka
and Volterra have shewn how to apply mathematics to evolutionary
biology, but which he just outside the scope of this book*.

An important element in a population, and one seldom easy to
define, is its age-composition. It may vary one way or the other;
for the diminution of a population may be due to a decrease in the
birth-rate, or to an increasing mortality among the old. A remark-
able instance is that of the food-fishes of the North Sea. Their
birth-rate is so high that the very young fishes remain, to all
appearance, as numerous as ever; those somewhat older are fewer
than before, and the old dwindle to a fraction of what they were
wont to be.

The rate of growth in other organisms

The rise and fall of growth-rate, the acceleration followed by
retardation which finds expression in the S^shaped curve, are seen
alike in the growth of a population and of an individual, and in
most things which have a beginning and an end. But the law of
large numbers smooths the population-curve; the individual Hfe
draws attention to its own ups and downs; and the characteristic
sigmoid curve is only seen in the simpler organisms, or in parts or
"phases" of the more complex lives. We see it at its simplest in
the simple growth-cycle, or single season, of an annual plant, which
cycle draws to its end at flowering; and here not only is the curve
simple, but its amplitude may sometimes be very large. The giant
Heracleum and certain tall varieties of Indian corn grow to twelve feet

* See (int. al.) A. J. Lotka, Elements of Physical Biology, Baltimore, 1925;
Theorie analytique des associations biologiques, Paris, 1934; Vito Volterra, Lemons
sur la theorie mathematique de la lutte pour la vie, 1931; Volterra et U. d'Ancona,
Les associations biologiques au point de vue mathematique, 1935; V. A. Kostitzin,
op. ci7.;\ etc

160

THE RATE OF GROWTH

[CH.

high in a summer; the kudzu vine (Pucraria) may grow twelve inches
in twenty-fouE hours, and some bamboos are said to have grown
twenty feet in three days (Figs. 32, 33).

10
Days

Fig. 32. Growth of Lupine. After Pfeffer.

Growth of Lupinus albus. (From G. Backman, after Pfeffer)

Length

Length

Day

(mm.)

DiflFerence

Day

(mm.)

DiflFerence

4

10-5

—

14

132-3

12-2

5

16-3'

5-8

15

140-6

8-3

6

23-3

7-0

16

149-7

9-1

7

32-5

9-2

17

155-6

5-9

8

42-2

9-7

18

158-1

2-5

9

58-7

14:5

19

160-6

2-5

10

77-9

19-2

20

161-4

0-8

n

93-7

15-8

21

161-6

0-2

12

107-4

13-7

13

1201

12-7

In the pre-natal growth of an infant the S-shaped curve is clearly
seen (Fig. 18); but immediately after birth another phase begins,
and a third is imphcit in the spurt of growth which precedes puberty.
In short, it is a common thing for one wave of growth (or cycle, as

Ill] IN VARIOUS ORGANISMS 161

some call it) to succeed another, whether at special epochs in a
lifetime, or as often as winter gives place to spring*.
20

Days
Fig. 33. Growth of Lupine: daily increments.

45 50
days

Fig. 34. Growth in weight of a mouse. After W. Ostwald.

In the accompanying curve of weight of the mouge (Fig. 34) we
see a slackening of the rate of growth when the mouse is about a
fortnight old, at which epoch it opens its eyes, and is weaned soon

* W. PfefiFer, Pflanzenphysiologie, 1881, Bd. ii, p. 78; A. Bennett, On the rate
of growth of the flower-stalk of Vallisneria spiralis and of Hyacinthus, Trans. Linn.
Soc. (2), I, Botany, pp. 133, 139, 1880; cited by G. Backman, Das Wachstums-

162 THE EATE OF GROWTH [ch.

after. At six weeks old there is another well-marked retardation;
it follows on a rapid spurt, and coincides with the epoch of puberty *.

In arthropod animals growth is apt to be especially discontinuous,
for their bodies are more or less closely confined until released by
the casting of the skin. The blowfly has its striking metamorphoses,
yet its growth is wellnigh continuous; for its larval skin is too thin
and dehcate to impede growth in the usual arthropod way. But
in a thick-skinned grasshopper or hard-shelled crab growth goes by
fits and starts, by steps and stairs, as Reaumur was the first to shew ;
for, speaking of insects f, he says: "Peut-etre est-il vrai generale-
ment que leur accroissement, ou au moins leur plus considerable
accroissement, ne se fait que dans le temps qu'ils muent, ou pendant
im temps assez court apres la mue. lis ne sont obHges de quitter
leur enveloppe que parce qu'elle ne prend pas un accroissement
proportionne a celui que prennent les parties qu'elle couvre."
All the visible growth of the lobster takes place once a year at
moulting-time, but he is growing in weight, more or less, all along.
He stores up material for months together; then comes a sudden
rush of water to the tissues, the carapace sphts asunder, the lobster
issues forth, devours his own exuviae, and lies, low for a month while
his new shell hardens.

The silkworm moults four 'times, about once a week, beginning
on the sixth or seventh day after hatching. There is an arrest or
retardation of growth before each moult, but our diagram (i^ig. 35)
is too small to shew the sHght ones which precede the first and

problem, in Ergebnisse d. Physiologie, xxxiii, pp. 883-973, 1931. These two cases
of Lupinus and Vallisneria, a^e among the many which lend themselves easily to
Backman's growth -formula, viz. Lupinus, \ogp= - 2-40 + 1-48 log T - 6-61 log^ T and
Vallisneria, log p = + 1-28 + 4-51 log T - 2-62 log^ T. See for an admirable resume
of facts, Wolfgang Ostwald, Ueher die zeitliche Eigenschaften der Entwicklungsvorgdnge
(71 pp.), 1908 (in Roux's Vortrdge, Heft v); and many later works.

* Cf. R. Robertson, Analysis of the growth of the white mouse into its con-
stituent processes, Journ. Gen. Physiology, vin, p. 463, 1926. Also Gustav
Backman, Wachstum d. w. Maus, Li^nds Univ. Arsskrift, xxxv, Nr. 12, 1939,
with copious bibliography. Backman analyses the complicated growth- curve of
the mouse into one main and three subordinate cycles, two df which are embryonic.
Cf. St Loup, Vitesse de croissance chez les souris. Bull. Soc. Zool. Fr. xviii,
p. 242, 1893; E. Le Breton and G. Schafer, Trav. Inst. Physiol. Strasburg, 1923;
E. C. MacDowell, Growth-curve of the suckling mouse. Science, Lxvin, p. 650,
1928; cf. Journ. Gen. Physiol, xi, p. 57, 1927; Ph. THeritier, Croissance. . .dfeins les
souris, Ann. Physiol, et Phys. Chemie, v, p. i, 1929.

I Memoires, iv, p. 191.

i

Ill]

OF CERTAIN INSECTS

163

second. Before entering on the pupal or chrysalis stage, when the
worm is about seven weeks old, a remarkable process of purgation

e

mgms

4000

1

3000

-

•

/

r

2000

-

^

1

\

1000

-

-

IV /

I

II

III /

1 , .1

10

15

20

25

30

35

40
days

Fig. 35. Growth in weight of silkworm. From Ostwald, after Luciani
and Lo Monaco.

takes place, with a sudden loss of water, and of weight, which
becomes the most marked feature of the curve* That the meta-

* Luciani e Lo Monaco, Arch. Ital. de Biologie, xxvii, p. 340, 1897; see also
Z. Kuwana, Statistics of the body-weight of the silkworm, Japan. Journ. Zool.
VII, pp. 311-346, 1937. Westwood, in 1838, quoted similar data from Count
Dandolo: according to whom 100 silkworms weigh on hatching 1 grain; after
the first four moults, 15^ 94, 270 and 1085 grains; and 9500 grains when full-grown.

164 THE RATE OF GROWTH [ch.

morphoses of an insect are but phases in a process of growth was
clearly recognised by Swammerdam, in the Bihlid Naturae*.

A stick-insect (Dexippus) moults six or seven times in as many
months ; it lengthens at every ilioult, and keeps of the same length
until the next. Weight is gained more evenly; but before each
moult the creature stops feeding for a day or two, and a little weight
is lost in the casting pf the skin. After its last moult the stick-
insect puts on more weight for a while; but growth soon draws to
an end, and the bodily energies turn towards reproduction.

We have careful measurements of the locust from moult to moult,
and know from these the relative growth-rates of its parts, though
we cannot plot these dimensions against time. Unlike the meta-
morphosis of the silkworm, the locust passes through five larval
stages (or "instars") all much ahke, ' until in a final moult the
"hoppers" become winged. Here are three sets of measurements,
of Hmbs and head, from stage to stage f.

Growth of locust, from one moult to another

Length (mm.) Percentage-grpwth Ratios

Anterior

Median

^

r

Anterior

Median

"*

Anterior

Median

"*

Stage

femur

femur

Head

femur

femur

Head

femur

femur

Heac

I

1-44

3-98

1-44

—

—

•^-

2-76

1-00

II

2-06

5-69

1-94

1-44

1-43

1-35

2-76

0-94

III

308

8-22

2-70

1-40

1-44

1-39

2-67

0-88

IV

4-53

11-94

3-71

1-47

1-45

1'37

2-76

0-82

V

6-40

17-22

4-89

1-41

1-44

1-32

2-69

0-76

Adult

8-03

22-85

5-59

1-25

1-33

V14

2-84

0-70

As a matter of fact the several parts tend to grow, for a time, at
a steady rate of compound interest, which rate is not identical for
head and hmbs, and tends in each case to fall off in the final moult,
when material has to be found for the wings. Some fifty years ago,
W. K. Brooks found the larva of a certain crab (Squilla) increasing
at each moult by a quarter of its own length ; and soon after
H. G. Dyar declared that caterpillars grow hkewise, from moult to
moult, by geometrical progression % . This tendency to a compoimd-

* 1737, pp. 6, 579, etc.

t A. J. Duarte, Growth of the migratory locust. Bull. Ent. Res. xxix, pp. 425-456,
1938.

% W. K. Brooks, Challenger Report on the Stomatopoda, 1886; H. G. Dyar;
Number of moult^ in lepidopterous larvae, Psyche, v, p. 424, 1896.

Ill] OF BROOKS'S LAW 165

interest rate in the growth and metamorphosis of insects is known
as Dyar's, sometimes as Brooks's, law. According to Przibram, an
insect moults as soon (roughly speaking) as cell-division has doubled
the number of cells throughout the larval body. That being so, each
stage or instar should weigh twice as much as the one before, and
each linear dimension should increase by ^2, or 1-26 times — a
Ineasure identical, to all intents and purposes, with Brooks's first
estimate. As sl first rough approximation the rule has a certain value.
According to Duarte's measurements the locust's total weight in-
creases from moult to moult by 2-31, 2-16, 242, 2-35, 2-21, or a
mean increase of 2-29, the cube-root of which is 1-32. Each phase
is doubled and more than doubled, in passing to the next*, but
Przibram's estimate is not far departed from.

Whatever truth Przibram's law may have in insects, or (as Fowler
asserted) in the Ostracods, it would seem to have none in the
Cladocera : and this for the sufficient reason that the shell (on which
the form of the creature depends) goes on growing all through post-
embryonic life without further division or multiphcation of its cells,
but only by their individual, and therefore collective, enlargementf.

Shells are easily weighed and measured and their various dimen-
sions have been often studied; only in oysters, pearl-oysters and
the Hke, have they been so kept under observation that their actual
age is known. The oyster-shell grows for a few weeks in spring just
before spawning time, and again in autumn when spawning is over ;
its growth is imperceptible at other times J.

* Cf. H. Przibram and F. Megusar, Wachstummessungen an Sphodromantis,
Arch. f. Entw. Mech. xxxiv, pp. 680-741, 1912; etc. How the discrepancy is
accounted for, by Bodenheimer and others, need not concern us here. But cf.
P.* P. Calvert, On rates of growth among. . .the Odonata, Proc. Amer. Phil. Soc.
Lxviii, pp. 227-274, 1929, who finds growth faster in nine cases out of ten than
Przibram's rule lays down.

Millet asserts, in support of Przibram's law, that in spiders mitotic cell-division
is confined to the epoch of the moult, and is then manifested throughout most of
the tissues (Bull, de Biologie {SuppL), viii, p. 1, 1926). On the other hand, the
rule is rejected by R. Gurney, Rate, of growth in Copepoda, Int. Rev. Hydrohiol.
XXI, pp. 189-27, 1929; Nobumasa Kagi, Growth-curves* of insect-larvae, Mem.
Coll. Agric. Kyoto, No. 1, 1926; and others.

t Cf. W. Rammer, Ueber die Giiltigkeit des Brooksschen Wachsturasgesetzes
bei den Cladoceren, Arch. f. Entw. Mech. cxxi, pp. 111-127, 1930.

X Cf. J. H. Orton, Rhythmic periods... in Ostrea, Jonrn. Mar. Biol. Assoc.
XV, pp. 365^27, 1928; Nature, March 2, 1935, p. 340.

166

THE RATE OF GROWTH

[CH.

The window-pane oyster in Ceylon (Placuna placenta) has been
kept under observation for eight years, during which it grows from
two inches long to six (Fig. 36). The young grow quickly, and slow
down asymptotically towards the end; an S-shaped beginning to the
growth-curve has not been seen, but would probably be found in
the growth of the first year. Changes of shape as growth goes on
are hard to see in this and other shells ; rather is it characteristic oi

Fig. 36.

12 3 4 5 6 7

Age in years

Growth of the window-pane oyster; short diameter of the shell.
From Pearson's data.

them to keep their shape from first to last unchanged. Nevertheless,
shght changes are there; in the window-pane oyster the shell grows
somewhat rounder; in seven or eight years the one diameter multi-
plies (roughly speaking) by eleven, and the other by ten*.

Window-pane oysters (Placuna)

Ratio

117
109
107
1-06
105

The American slipper-limpet has lately and quickly become a pest
on English oyster-beds. Its mode of growth is interesting, though

* Joseph Pearson, The growth-rate. . .oi Placuna placenta, Ceylon Bulletin, 1928.

Short

Long

diameter

diameter

(mm.)

(mm.)

150

17-6

650

70-5

102-5

109-7

132-5

139-9

167-5

175-2

Ill

OF THE GROWTH OF SHELLFISH

167

the actual rate remains unknown. It grows a little longer and
narrower with age. Its weight-length coefficient (of which we shall
have more to say presently) increases as time goes on, and appears
to follow a wavy course which might be accounted for if the
shell grew thinner and then thicker again, as if ever so little more
lime were secreted at one season than another. The growth of a
shell, or the deposition of its calcium carbonate, is much influenced
by temperature; clams and oysters enlarge their shells only so long
as the temperature stands above a certain specific minimum, and
the mean size of the same hmpet is very different in Essex and in
the United States*. Curious peculiarities of growth have been
discovered in slipper-hmpets. Young limpets clustered roimd an
old female grow slower than others which Uve sohtary and apart.
The solitary forms become in turn male, hermaphrodite and at last
female, but the gregarious or clustered forms . develop into males,
and so remain; development of male characters and duration of
the male phase depend on the presence or absence of a female in
the near neighbourhood.

Measurements of slipper-limpets
{From J. H. Eraser's data, epitomised)

No.

Mean length

Breadth

Ratio

Weight

measured

(mm.)

(mm.)

L/B

(gra-)

WjD

3

15-3

8-8

1-74

0-33

92

8

17-6

9-8

1-80

0-46

84

9

19-4

10-5

1-85

.0-63

88

16

21-5

11-5

1-87

0-77

77

18

23-5

125

1-88

104

80

41

25-5

13-7

1-86

1-37

85

91

27-4

14-5

1-89

1-81

88

125

39-4

15-4

1-91

2-33

92

98

31-4

16-5

1-90

3-22

104

70

33-6

17-8

1-89

3-61

95

38

35-5

18-6

1-90

4-28

95

10

37-3

19-5

1-91

4-95

95

1

321

19-4

201

5-35

90

Mean 1-87

89-3

* Cf. J. H. Fraser, On the size of Urosalpinx etc., Proc. Malacol. Soc. xix,
pp. 243-254, 1931. Much else is known about the growth of various limpets,
their seasonal periodicities, the change of shape in certain species, and other
matters; cf. E. S. Russell, Growth of Patella, P.Z.S. cxcix, pp. 235-253; J. H.
Orton, Journ. Mar. BioL Assoc, xv, pp. 277-288, 1929; Noboru Abe, Sci. Rep.
Tohoku Imp. Univ. Biol, vi, pp. 347-363, 1932, and Okuso Hamai, ibid, xii,
pp. 71-95, 1937.

168

THE RATE OF GROWTH

[CH.

The growth of the tadpole * is Hkewise marked by epochs of
retardation, and finally by a sudden and drastic change (Fig. 37).
There is a shght diminuti9n in weight immediately after the little
larva frees itself from what remains of the egg ; there is a retardation

mgms.

1

6000

'

/

')

5000
4000

~ •

/ ^

/ 1

j

: 1
■j j

3000

-

/

i

i

2000

-

1

/

1000

- 1

u3

J

10 20 30 40 50 60 70 80 90

days

Fig. 37. Growth in weight of tadpole. From Ostwald, after Schaper.

of growth about ten days later, when the external gills disappear; and
finally the complete metamorphosis, with the loss of the tail, the growth
of the legs and the end of branchial respiration, brings about a loss
of weight amounting to wellnigh half the weight of the full-grown

* Cf. (int. at.) Barfurth, Versiiche iiber die Verwandlung der Froschlarven
Arch. f. mikroak. Anat. xxix, 1887.*

Ill]

OF LARVAL EELS

169

Fig. 38.

Development of eel : from Leptocephalus larvae to young elver.
After Johannes Schmidt.

170 THE RATE OF GROWTH [ch-

larva. At the root of the matter hes the simple fact that meta-
morphosis involves wastage of tissue, increase of oxidation, expendi-
ture of energy and the doing of work. While as a general rule the
better the animals be fed the quicker they grow and the sooner they
metamorphose, Barfurth has pointed out the curious fact that a
short spell of starvation, just before metamorphosis is due, appears
to hasten the change.

The negative growth, or actual loss of bulk and weight which
often, and perhaps always, accompanies metamorphosis, is well
shewn in the case of the eel *. The contrast of size is great between
the flattened, lancet-shaped Leptocephalus larva and the httle black,
cylindrical, almost thread-Uke elver, whose magnitude is less than
that of the Leptocephalus in every dimension, even at first in length
(Fig. 38), as Grassi was the first to shew.

The lamprey's case is hardly less remarkable. The larval or
Ammocoete stage lasts for three years or more, and metamorphosis,
though preceded by a spurt of growth, is followed by an actual
decrease in size. The little brook lamprey neither feeds nor grows
after metamorphosis, but spawns a few months later and then dies;
but the big sea-lampreys become semi-parasitic on other fishes, and
live and grow to an unknown agef.

Such fluctuations as these are part and parcel of the general flux
of physiological activity, and suggest a finite stock of energy to be
spent, now more now less, on growth and other modes of expenditure.
The larger fluctuations are special interruptions in a process which
is never continuous, but is perpetually varied by rhythms of various
kinds and orders. Hofmeister shewed long ago, for instance, that
Spirogyra grows by fits and starts, in periods of activity and rest
alternating with one another at intervals of so many minutes {
(Fig. 39). And Bose tells us that plant-growth proceeds by tiny and
perfectly rhythmical pulsations, at intervals of a few seconds of time.

* Johannes Schmidt, Contributions to the life-history of the eel, Rapports
du Conseil Intern, pour V exploration de la mer,y, pp. 137-274, Copenhagen, 1906;
and other papers.

t Cf. {int. al.) A. Meek, The lampreys of the Tyne, Rep. Dove Marine Laboratory
(N.S.), VI, p. 49, 1917; cf. L. Hubbs, in Papers of the Michigan Academy, iv,
p. 587, 1924.

% Die Lehre der Pflanzenzelle, 1867. Cf. W. J. Koningsberger, Tropismus und
Wachstum (Thesis), Utrecht, 1922.

Ill]

OF PERIODIC GROWTH

171

A crocus grows, he says, by little jerks, each with an amphtude of
about 0-002 mm., every twenty seconds or so, each increment being
followed by a partial recoil* (Fig. 40). If this be so we have come

20 40 60 80 100 120 140 160 180 200 220 240 260

minutes

Fig. 39. Growth in length (mm.) of Spirogyra. From Ostwald, after Hofmeister.

Fig. 40.

seconds.

Pulsations of growth in Crocus, in micro-millimeter?
After Bose.

down, so to speak, from a principle of continuity to a principle of dis-
continuity, and are face to face with what we might call, by rough
analogy, "quanta of growth." We seem to be in touch with things
of another order than the subject of this bookt.

* J. C. Bose, Plant Response, 1906, p. 417; Growth and Tropic Movements of
Plants, 1929.

t There is an apparent and perhaps a real analogy between these periodic
phenomena of growth and the well-known phenomenon of periodic, or oscillatory,
chemical change, as described by W. Ostwald and others; cf. (e.g.) Zeitschr. f.
phys. Chem. xxxv, pp. 33, 204, 1900.

172 THE RATE OF GROWTH [ch.

We may want now and then to make use of scanty data, and find
a rougli estimate better than none. The giant tortoises of the
Galapagos and the Seychelles grow to a great age, and some have
weighed 5001b. and more; but the scanty records of captive
tortoises shew much variation, depending on food and climate as
well as age, Ninety young tortoises brought from the Galapagos
in 1928 to the southern United States weighed on the average
18J lb., and grew to 44-3 lb. in two years. Six taken to Honolulu
weighed 26 J lb. each in 1929, and 63 lb. each the following year.
Another, kept in CaHfornia, weighed 29 lb. and 360 lb. seven years
later, but only gained 65 lb. more in the next seven years. Growth,

1899 1906 1913

Fjg. 41. Approximate growth in weight of Galapagos tortoise.

as usual, is quick to begin with, slower lat^r on, and in the old giants
must be slow indeed. If we plot (Fig. 41 ) the three successive weights
of the CaHfornian specimen, at first they help us little; but we can
fit an S-shaped curve to the three points as a first approximation,
and it suggests, with some plausibility, that, at 29 lb. weight the
tortoise was from two to three years old. A loggerhead turtle,
which reaches a great size, was found to grow from a few grammes
to 42 lb. in three years, and to double that weight in another year
and a half; these scanty data are in fair accord, ^o far as they go,
with those for the giant tortoises*.

* For these and other data, see C. H. Townsend, Growth and age in the giant
tortoises of the Galapagos, Zoologica, ix, pp. 459-466, 1931; G. H. Parker, Growth
of the loggerhead turtle, Amer. Naturalist, lxvii, pp. 367-373, 1929; Stanley F.
Flower, Duration of Life in Animals, in, Reptiles, P.Z.S. (A), 1937, pp. 1-39.

Ill]

OF WHALES AND TORTOISES

173

The horny plates of the tortoise grow, to begin with, a trifle faster
than the bony carapace below, and are consequently wrinkled into
folds. There is some evidence, at least in the young tortoises, that
these folds come once a year, which is as much as to say that there
is one season of the year when the growth-rates of bony and horny
carapace are especially discrepant. This would give an easy estimate
of age; but it is plainer in some species than in others, and it never
lasts for long.

re m years
Fig. 42. Growth-rate (approximate) of blue and finner whales.

The blue whale, or ' Sibbald's rorqual, largest of all animals,
grows to 100 ft. long or thereby, the females being a httle bigger
than the males. The mother goes with young eleven months. The
calf measures 22 to 25 ft. at birth, and weighs between three and
four tons; it is bom big, were it smaller it might lose heat too
quickly. It is weaned about nine months later, and is said to be
some 16 metres, or say 53 ft., long by then. It is believed to be
mature at two years old, by which time it is variously stated to be
60 or even 75 ft. long ; the modal size of pregnant females is about
80 ft. or rather more. How long, the whale takes to grow the
further 15 or 20 feet which bring it to its full size is not known;
but, even so far, the rapid growth and early maturity seem very
remarkable (Fig. 42). The Norwegian whalers give us statistics,

174

THE RATE OF GROWTH

[CH.

month by month during the Antarctic season, of the sizes of pregnant
females and the foetuses they contain; and from these I draw the
following averages:

Antarctic blue whales; length of mother and of foetus
(Season 1938-39)

Number
measured

Mother

Foetus

Nov.

1, 1938
16

59
86

84-0 ft.
83-0

4-2 ft
4-6

Dec.

1
16

359
522

84-0
83-6

61

7-0

Jan.

1, 1939
16

403
317

83-7
84-8

8-4
9-3

Feb.

1
16

184
125

83-9
83-9

11-2'
12-3

Mar.

1

71

83-6

14-5

2126

83-8

^ .-

NOV

DEC

JAN

FEB

MAR

Fig. 43. Pre-natal growth of blue whale. Average monthly sizes,
from data in International Whaling Statistics, xiv, 1940.

The observations are rough but numerous. At the lower end of
the scale measurements are few, and the value indicated is probably
too high; but on the whole the curve of growth taUies with other
estimates, and points to birth about June or July, and to conception
about the same time last year (Fig. 43). The mean size of the
mother- whales does not alter during the five months in question;

Ill] THE GROWTH OF WHALES 175

they do not seem to be increasing, though at 84 ft. they still have
another 10 feet or more to grow. They may grow slower, and live
longer, than is often supposed*.

On the other hand, if we draw from the same official statistics
the mean size of mother-whale and foetus at some given epoch of
the year (e.g. March 1934), there appears to be a marked correlation
between them, such as would indicate very considerable growth
of the mother during the months of pregnancy. The matter deserves
further study, and the data need confirmation.

Blue whales; length of mother and foetus {March 1934)

Size of

Size (ft.)

foetus (ft.)

Number
observed

X

smoothed
in threes

Mother

Foetus

1

74

10

' —

1

75

7-0

4-4

5

76

5-2

6-4

7

77

7-3

6-4

9

78

6-7

6-9

10

79

6-7

6-8

21

80

71

7-2

27

81

7-7

7-5

28

, 82

7-6

7-9

33

83

'8-4

8-3

38

84

8-9

8-6

46

85

8-5

8-6

37

86

8-5

8-8

19

87

9-5

9-3

18

88

9-9

10-3

12

89

11-4

10-9

18

90

11-5

111

9

91

10-4

111

2

92

11-5

—

341

On the growth of fishes, and the determination of their age

We may. keep a child under observation, and weigh and measure
him every day; but more roundabout ways are needed to determine
the age and growth of the fish in the sea. A few fish may be caught
and marked, on the chance of their being caught again; or a few

* The growth of the finner whale, or common rorqual, is estimated as follows
(Hamburg Museum) : at birth, 6 m.; at 6 months, 12 m. ; at one and two years old,
15 and 19 m. ; when full-grown, at 6-8 ( ?) years old, 21 m. For data, see Hvalradets
Skrifter and Jhtemational Whaling Statistics, passim ; also N. Mackintosh and others
in Discovery Reports; also Sigmund Rusting, Statistics of whales and whale-
foetuses, Rapports du Conseil Int. 1928; etc.

176 THE RATE OF GROWTH [ch.

more may be kept in a tank or pond and watched as they grow.
Both ways are slow and difficult. The advantage of large numbers
is not obtained ; and it is needed all the more because the rate of
growth turns out to be very variable in fishes, as it doubtless is
in all cold-blooded or " poecilothermic " animals: changing and
fluctuating not only with age and season, but with food-supply,
temperature and other known and unknown conditions. Trout in
a chalk-stream so differ from those in the peaty water of a highland
burn that the former may grow to three pounds weight while the
latter only reach four ounces, at three years old or four*.

It is found (and easily verified) that shells on the seashore, kind
for kind, do not follow normal curves of frequency in respect of
magnitude, but fall into size-groups with intervals between, so
constituting a multimodal curve. The reason is that they are not
born all the year round, as we are, but each at a certain annual
breeding-season ; so that the whole population consists of so many
"groups," each one year older, and bigger in proportion, than
another. In short we find size-groups, and recognise them as age-
groups. Each group has its own spread or scatter, which increases
with ^ize and age; even from the first one group tends to overlap
another, but the older groups do so more and more, for they have
had more time and chance to vary. Hence this way of determining
age gets harder and less certain as the years go by; but it is a safe
and useful method for short-lived animals, or in the early lifetime
of the rest. Aristotle's fishermen used it when they recognised
three sorts or sizes of tunnies, the auxids, pelamyds and full-grown
fi^h; and when they found a scarcity of pelamyds in one year to
be followed by a failure of the tunny-fishery in the nextf.

Shells lend themselves to this method, as Louis Agassiz found when
he gathered periwinkles on the New England shore. Winckworth
found the Paphiae in Madras harbour "of two sizes, one group just
under 15 mm. in length, the other nearly all over 30 mm. A small
sample, dredged ^ve months earlier hora the same ground, was inter-
mediate between the other two." When the mean sizes of the two
groups were plotted against time, the lesser group being shifted

* Cf. C. A. Wingfield, Effect of environmental factors on the growth of brown
trout, Journ. Exp. Biol, xvii, pp. 435-448, 1939.
I Aristotle, Hist. Anim. vi, 571 a.

Ill

THE GROWTH OF FISHES 177

back a year, a growth-curve extending over two seasons was obtained ;
when extrapolated, it seemed to start from zero about May or June,
and this date, at the beginning of the hot season, was in all proba-
bihty the actual spawning time. Growth stopped in winter, a
common thing in our northern climate but surprising at Madras,
where the sea-temperature seldom falls below 24° C. Shells over
40 mm. long were rare, and over 50 mm. hardly to be found — an
indication that Paphia seldom lives over a third season. Here then,
though the numbers studied were all too few, the method tells us
with httle doubt or ambiguity the age of a sample and the growth-
rate of the species to which it belongs*.

Dr C. J. G. Petersen of Copenhagen brought this method into use
for the study of fishes, and up to a certain point it is safe and
trustworthy though seldom easy. For one thing, it is hard to get
a "random sample" of fish, for one net catches the big and another
the small. The trawl-net takes all the big, but lets more and more
of the small ones through. The drift-net catches herring by their
heads; if too big, the head fails to catch and the fish goes free, if
too small the fish shps through; so the net selects a certain modal
size according to its mesh, and with no great spread or scatter.
When we use Petersen's method and plot the sizes of our catch of
fish, the younger age-groups are easily recognised, even though they
tend to overlap; but the older fish are few, each size-group has a
wider spread, and soon the groups merge together and the modal
cusps cease to be recognisable. There is no way, save a rough
conjectural one, of analysing the composite curve into the several
groups of which it is composed; in short, this method works well
for the younger, but fails for the older fish.

Fig. 44 is drawn from a catch of some 500 small cod, or codhng,
caught one November in the Firth of Forth, in a small-meshed
experimental trawl-net. They are too few for the law of large
numbers to take full effect; but after smoothing the curve, three
peaks are clearly seen, with some sign of a fourth, indicating about

* R. Winckworth, Growth of Paphia undulata, Proc. Malacolog. Soc. xix,
pp. 171-174, 1931. Cf. {int. al.) Weymouth, on Mactra stultorum. Bull. Calif.
Fish Comm. vii, 1923; Orton, on Cardium, Journ. Mar. Biol. Assoc, xiv, 1927,
on Ostrea, and on Patella, ibid, xv, 1928; Ikuso Hamai, on Limpets, Sci. Rep.
Tohoku Imp. Univ. (4), xn, 1937.

178 THE RATE OF GROWTH [ch.

11cm., 26, 44 and 60 cm., as the mean or modal sizes of four
successive broods. The dwindhng heights of the successive cusps
are a first approximation to a "curve of mortahty," shewing how
the young are many and the old are few. Again, plotting the several
sizes against time, we should get our curve of growth for four years,
or a first rough approximation to it. Thus we learn from a random
sample, caught in a single haul, the mean (or modal) sizes of a fish
at several epochs of its fife, say at two, three or even more successive
intervals of a year ; and we learn (to a first approximation) its rate
of growth and its actual age, for the slope of the growth-curve,
drawing to the base-fine, points to the time when growth began.

^, Cod. Nov. 1906. F. of F

20 JO 40

Length, in centimetres
Fig. 44. A catch of cod, shewing a multimodal curve of frequency.

Another haul, soon after, will add new points to the curve, and
confirm our first rough approximation.

An experiment in the Moray Firth, a month or two later, shewed
the first three annual groups in much the same way; but it also
shewed another group, of about 90 cm. long, and others larger still.
At first sight these did not seem to fit on to our four successive
year-groups, of 11, 26, 44 and 60 .cm.; but they did so after all,
only with a gap between. They were older fish, six and seven years
old, which had come back to the Moray Firth to breed after spending
a couple of years elsewhere.

It was thought at first that every such experiment should tally
with another, and bring us to a more and more accurate knowledge
of the growth-rate of this fish or that; but there were continual
discrepancies, and it was soon found that the rate varied from place

Ill]

THE GROAVTH OF FISHES

179

to place, from month to month, and from one year to another.
The growth-rate of a fish varies far more than does that of a warm-
blooded animal. The general character of the curve remains*,
save that the fish continues to grow even in extreme old age, but
it draws towards its upper asymptote with exceeding slowness.

Fig. 45. Growth of cod (after Michael Graham); and of
mullet (after C. D. Serbetis).

The following estimate of the mean growth of North Sea cod is
based, by Michael Graham, on a great mass of various evidence;
and beside it, for comparison, is an estimate for the grey mullet,
by C. D. Serbetis. The shape of the curve (Fig. 45) is enough to
indicate that at six years old the cod is still growing vigorously f,
while the grey mullet has all but ceased to grow. As a matter of

* It is essentially an S-shaped curve, as usual; but the conditions of larval life
obscure the first beginnings of the S.

t Norwegian results, based largely on otoliths, are different. Gunner Rollefsen
holds that the spawning cod, or skrei, do not reach maturity, for the most part,
till 10 or 11 years old, and grow by no more than 1 to 3 cms. a year (Fiskeriskrifter,
Bergen, 1933).

180 THE RATE OF GROWTH [ch.

fact, 90 cm. is, or was till lately, the median size of cod* in our
Scottish trawl-fishery; one-tenth are over a metre long and the
largest are in the neighbourhood of 120 cm., with an occasional
giant of 150 cm. or even more. But it has come to pass that fish
of outstanding size are seen no more save on the virgin fishing
grounds; a Greenland halibut, brought home to Hull in 1938,
weighed four hundredweight, was nearly two feet thick, and must
have been of prodigious age.

Age (years)

1

2

3

4

5

6

Length of cod (cm.)

18

36

55

68

79

89

Length of grey mullet

21

36

46

51

53

55

There are other ways of determining, or estimating, a fish's age.
The Greek fishermen shewed Aristotle f how to tell the age of the
purple Murex, up to six years old, by counting the whorls and
sculptured ridges of the shell, and also how to estimate the age of
a scaly fish by the size and hardness of its scales ; and Leeuwenhoek
saw that a carp's scales J bear concentric rings, which increase in
number as the fish grows old. In these and other cases, as in the
woody rings of a tree, some part of plant or animal carries a record
of its own age; and this record may. be plain and certain, or may
too often be dubious and equivocal.

The scales of most fishes shew concentric rings, sometimes (as in
the herring) of a simple kind, sometimes (as in the cod) in a more
complex pattern; and the ear-bones, or otohths, shew opaque
concentric zones in their translucent structure. The scales are
''read" with apparent ease in herring, haddock, salmon, the otohths
in plaice and hake; but the whole matter. is beset with difficulties,
and every result deserves to be checked and scrutinised §.

* As distinguished from "codling."

t Hist. Animalium, 5476, 10; 6076, 30.

X The carp-breeder is especially interested in the age of his fish; for, like the
brewer with his yeast, his profit depends on the rate at which thej'^ grow.
Leeuwenhoek 's and other early observations were brought to light by C. HoflFbauer,
Die Alterbestimmung der Karpfen an seiner Schuppen, Jahresber. d. schles.
Fischerei-Vereins, Breslau, 1899.

§ Thus, for instance, Mr A. Dannevig says (On the age and growth of the cod,
Fiakeridirektorets Skrifter, 1933, p. 82): "as to the problem of the determination
of the age of the cod by means of scales and otohths, all workers agree that the
method is useful. But on a number of fundamental points there are just as many
divergences of opinion as there are investigators."

Ill] OF THE SIZE AND AGE OF HERRING 181

In the following table, we see (a) the sizes, and (6) the number

of scale-rings, in a sample of some 550 herring from the autumn
fishery off the east of Scotland.

Rings 3 4 5 6 7 8 9 10 11 12 Total Mean

cm. rings

31 — — — 1 1 _ 1 _ _ 1 4 8-5

30 — — — 7 5 6 4 — — — 22 7-3

29 — — 5 18 13 6 6 1 1 1 51 70

28 — 3 29 38 11 3 3 1 — — 88 5-9

27 2 13 41 34 5 5 2 _ _ _ 102 5-1

26 7 43 64 29 — — 1 _ _ _ 144 5.0

25 4 36 41 11 — — — — — — 92 4-6

24 2 17 15 4 — — — — — — 38 4-8

23 — 5 — — — — — — — — 5 4K)

Total 15 117 195 142 35 20 17 2 1 2

Mean 25-6 25-4 26-5 27-4 28-6 28-7 28-8 28-5 — —

In this sample, the sizes of the 550 fish are grouped in a somewhat
skew curve, about a mode at 26 cm. ; and the numbers of scale-rings
group themselves in like manner, but with rather more skewness,
about a modal number of five rings. Either way we look at it,
there is only one "group" of fish; and it is highly characteristic
of the herring that a single sample, taken from a single shoal,
exhibits a unimodal curve. Accepting in principle the view that
scale-rings tend to synchronise with age in years, we may draw this
first deduction that our sample consists in part (if not in whole)
of five-year old fish, whose average length is about 26 cm. ; and
this length, of 26 cm. for 5-ringed, or 5-year-old herring, agrees well
with many other determinations from the same region. We shall
be on the safe side if we deal, after this fashion, with the one
predominant group, or mode, in each sample of fish; and Fig. 46
shews an approximate curve of growth for our East Coast herring
drawn in this way.

But the further assumption is commonly and all but universally
made that each individual herring carries the record of its age on its
scale-rings. If this be so, then our sample of 550 fish is a com-
posite population of some ten separate broods or successive ages,
all mixed up in a shoal. And again, if so, the 5-year-olds in the
said population average 26-5 cm. in length, the 3-year-olds 25-6 cm.,
the 10-year-olds 28-5 cm. ; but these values do not fit into a normal

182

THE RATE OF GROWTH

[CH.

curve of growth by any means. Still more obvious is it that the
several year-classes (if such they be) do not tally with the age-
composition of any ordinary population, nor agree with any ordinary
curve of mortality. But even if we had ten separate year-groups
represented here, which I most gravely doubt, all that we know of
the selective action of the drift-net forbids us to assume that we
are deahng with a fair random sample of the herring population;
so that, even though the number of rings did enable us to distinguish
the successive broods, we should still have no right to assume that

Fig. 46.

1 2 3 4 5 6

Years
Mean curve of growth of Scottish (East Coast) herring.

these annual broods actually combine in the proportions shewn,
to form the composite population.

It is held by many (in the first" instance by Einar Lea) that we
may deduce the dimensions of a herring at each stage of its past
life from the corresponding dimensions of the rings upon its scales.
Some such relation nmst obviously exist, but it is an approximation
of the roughest kind. For it involves the assumption not only that
the scales add ring to ring regularly year to year, and that fish
and scale grow all the while at corresponding rates or in direct
proportion to one another, but also that the scale grows by mere

Ill] OF THE SIZE AND AGE OF HERRING 183

accretion, each annual increment persisting without further change
after it is once laid down. This is what happens in a moUuscan
shell, which is secreted or deposited as mere dead substance or
"formed material"; but it is by no means the case in bone, and
we have httle reason to expect it bf the bony mesoblastic tissue of
a fish's scale. It is much more likely (though we do not know for
sure) that "osteoblasts" and "osteoclasts" continue (as in bone) to
play their part in the scale's growth and maintenance, and that
some sort of give and take goes on. In any case, it is a matter of

Mean ajyparent length of one-year-old herring, as deduced by
scale-reading from herring of various ages or ''year-classes'^''

Year-class (or number
of rings)

.>

3

4

.■)

6

7

8

9

Estimated length at
1 year old

140

13-2

12-7

I2-0

12- 1

11-8

11-9

11-8

fact and observation that the rings alter in breadth as the fish goes
on growing f ; that the oldest or innermost rings grow steadily
narrower, while the outermost hardly change or even widen a little ;
that- the relative breadths of successive rings alter accordingly;
and it follows that when we try to trace the growth of a herring
through its lifetime from its scales when it is old, the result is more
or less misleading, and the values for the earher years are apt to
be much too small. The whole subject is very difficult, as we might
well expect it to be; and, I am only concerned to shew some
small part of its difficulty J.

While careful observations on the rate of growth of the higher
animals are scanty, they shew so fax as they go that the general
features of the phenomenon are much the same. Whether the
animal be long-lived, as man or elephant, or short-lived hke horse §

* From T. Emrys Watkin, The Drift Herring of the S.E. of Ireland, Rapports du
Conseil pour V Exploration de la Mer, lxxxiv, p. 85, 1933.

t Cf. {int. al.) Rosa M. Lee, Methods of age and growth determination in fishes
by means of scales, Fishery Investigations, Dept. of Agr. and Fisheries, 1^20.

X The copious literature of the subject is epitomised, so far, by Michael Graham,
in Fishery lyivestigations (2), xi, No. 3, 1928.

§ There is a famous passage in Lucretius (v, 883) where he compares the course
of life, or rate of growth, in the horse and his boyish master: Principio circum
tribus actis impiger annis Floret equus, puer hautquaquam, etc.

184 THE RATE OF GROWTH [ch.

or dog, it passes through the same phases of growth ; and, to quote
Dr Johnson again, "whatsoever is formed for long duration arrives
slowly to its maturity*." In all cases growth begins slowly; it
attains a maximum velocity somewhat early in its course, and
afterwards slows down (subject to temporary accelerations) towards
a point where growth ceases altogether. But in cold-blooded
animals, as fish or tortoises, the slowing down is greatly protracted,
and the size of the creature would seem never to reach, but only
to approach asymptotically, to a maximal limit. This, after all,
is an important difference. Among certain still lower animals
growth ceases early but Hfe goes on, and draws (apparently) to no
predetermined end. So sea-anemones have been kept in captivity
for sixty or even eighty years, have fed, flourished and borne
offspring all the while, but have shewn no growth at all.

The rate of growth of various parts or organs f

That the several parts and organs of the body, within and
without, have their own rates of growth can be amply demonstrated
in the case of man, and illustrated also, but chiefly in regard to
external form, in other animals. There lies herein an endless
field for the study of correlation and of variability J.

In the accompanying table I show, from some of Vierordt's data,
the relative weights at various ages, compared with the weight at
birth, of the entire body, and of brain, heart and liver; also the
changing relation which each of these organs consequently bears,
as time goes on, to the weight of the whole body (Fig. 47) §.

* All of which is tantamount to a mere change of scale of the time-curve.

f This phenomenon, of incrementum inequale, as opposed to incrementum in
universum, was most carefully studied by Haller: "Incrementum inequale multis
modis fit, ut aliae partes corporis aliis celerius increscant. Diximus hepar minus
fieri, majorem pulmonem, minimum thymum, etc." (Elem. viii (2), p. 34.)

X See {int. al.) A. Fischel, Variabilitat und Wachsthum des embryonalen
Korpers, Morphol. Jahrb. xxiv, pp. 369-404, 1896; Oppel, Vergleickung des
Entwickelungsgrades der Organe zu verschiedenen Entwickelungszeiten hei Wirhel-
thieren, Jena, 1891; C. M. Jackson, Pre-natal growth of the human body and the
relative growth of the various organs and parts, Amer. Journ. of Anat. ix, 1909;
and of the albino rat, ibid, xv, 1913; L. A. Calkins, Growth of the human body in
the foetal period. Rep. Amer. Assoc. Anat. 1921. For still more detailed measure-
ments, see A. Arnold, Korperuntersuchungen an 1656 Leipziger Studenten, Ztschr.
f. Konstitutionslehre, xv, pp. 43-113, 1929.

§ From Vierordt's Anatomische Tabellen, pp. 38, 39, much abbreviated.

Ill]

OF PARTS OR ORGANS

185

Weight of various organs, compared with the total weight of the human
body' (male). (From Vierordfs Anatomische Tahellen)

Percentage increase

Percentage of body-wt.

Wt.

r

-A. , , .

^

,

A

s

Age

(kgm.)

Body

Brain

Heart

Liver

Brain

Heart

Live

31

10

10

10

10

12-3

0-76

4-6

1

9-0

2-9

2-5

1-8

2-4

10-5

0-46

3-7

2

110

3-6

2-7

2-2

30

9-3

0-47

3-9

3

125

40

2-9

2-8

3-4

8-9

0-52

3-9

4

140

4-5

3-5

31

4-2

9-5

0-53

4-2

5

15-9

51

3-3

3-9

3-8

7-9

0-51

3-4

6

17-8

5-7

3-6

3-6

4-3

7-6

0-48

3-5

7

19-7

6-4

3-5

3-9

4-9

6-8

0-47

3-5

8

21-6

70

3-6

40

4-6

6-4

0-44

30

9

23-5

7-6

3-7

4-6

50

61

0-46

30

10

25-2

8-1

3-7

5-4

5-9

5-6

0-51

3-3

11

27-0

8-7

3-6

6-0

61

50

0-52

3-2

12

290

9-4

3-8

(4.1)

6-2

4-9

(0-34)

30

13

331

10-7

3-9

7-0

7-3

45

0-50

31

14

371

120

3-4

9-2

8-4

3-5

0-58

3-2

15

41-2

13-3

3-9

8-5

9-2

3-6

0-48

32

16

45-9

14-8

3-8

9-8

9-5

3-2

0-51

30

17

49-7

160

3-7

10-6

10-5

2-8

0-51

30

18

53-9

17-4

3-7

10-3

10-7

2-6

0-46

2-8

19

57-6

18-6

3-7

11-4

11-6

2-4

0-51

2-9

20

59-5

19-2

3-8

12-9

110

2-4

0-51

2-6

21

61-2

19-7

3-7

12-5

11-5

2-3

0-49

2-7

22

62-9

20-3

3o

13-2

11-8

2-2

0-50

2-7

23

64-5

20-8

3-6

12-4

10-8

2-2

0-46

2-4

24

3-7

131

13-0

—

—

—

25

66-2

21-4

3-8

12-7

12-8

2-2

0-46

2-8

Fig. 47.

10

Age in years
Relative growth in weight of brain, heart and body of man.
From Quetelet's data (smoothed curves).

186 THE RATE OF GROWTH [ch.

We see that neither brain, heart nor Hver keeps pace by any
means with the growing weight of the whole; there must then
be other parts of the fabric, probably the muscles and the bones,
which increase more rapidly than the general average. Heart and
liver grow nearly at the same rate, the liver keeping a little ahead
to begin with, and the heart making up on it in the end; by the
age of twenty-five both have multiplied their original weight at
birth about thirteen times, but the body as a whole has multiplied
by twenty-one. In contrast to these the brain has only multiplied
its weight about three and three-quarter times, and shews but little
increase since the child was four or five, and hardly any since it
was eight years old. Man and the gorilla are born with brains much
of a size; but the gorilla's brain stops growing very soon indeed,
while the child's has four years of steady increase. The child's
brain grows quicker than the gorilla's, but the great ape's body
grows much quicker than the child's; at four years old the young
gorilla has reached about 80 per cent, of his bodily stature, and the
child's brain has reached about 80 per cent, of its full size.

Even during foetal life, as well as afterwards, the relative weight of the
brain keeps on declining. It is about 18 per cent, of the body- weight in the
third month, 16 per cent, in the fourth, 14 per cent, in the fifth; and the
ratio falls slowly till it comes to about 12 per cent, at birth, say 10 per cent,
a year afterwards, and little more than 2 per cent, at twenty*. Many statistics
indicate a further decrease of brain-weight, actual as well as relative. The
fact has been doubted and denied ; but Raymond Pearl has shewn evidence
of a slow decline continuing throughout adult life f.

The latter part of the table shews the decreasing weights of the
organs compared with the body as a whole: brain, which was
12 per cent, of the body- weight at birth, falling to 2 per cent, at
five-and-twenty ; heart from 0-76 to 0-46 per cent.; liver from
4-6 to 2-78 per cent. The thyroid gland (as we know it in the rat)
grows for a few weeks, and then diminishes during all the rest of
the creature's Hfetime; even during the brief period of its own
growth it is growing slower than the body as a whole.

It is plain, then, that there is no simple and direct relation, holding

*■ Cf. J. Ariens Kappers, Proc. K. Akad. Wetensch., Amsterdam, xxxix, No. 7, 1936.
t R. Pearl, Variation and correlation in brain-weight, Biometrika, iv, pp. 13-104,
1905.

Ill] OF PARTS OR ORGANS 187

good throughout life, between the size of the body and its organs;
and the ratio of magnitude tends to change not only as the individual
grows, but also with change of bodily size from one individual, one
race, one species to another. In giant and pigmy breeds of rabbits,
the organs have by no means the same ratio to the body- weight ; but
if we choose individuals of the same weight, then the ratios tend to
be identical, irrespective of breed*. The larger breeds of dogs are
for the most part lighter and slenderer than the small, and the organs
change their proportions with their size. The spleen keeps pace
with the weight of the body ; but the liver, hke the brain, becomes
relatively less. It falls from about 6 per cent, of the body-weight
in little dogs to rather over 2 per cent, in a great hound f.

The changing ratio with increasing magnitude is especially
marked in the case of the brain, which constitutes (as we have just
seen) an eighth of the body- weight at birth, and but one-fiftieth at
twenty-five. This faUing ratio finds its parallel in comparative
anatomy, in the general law that the larger the animal the smaller
(relatively) is the brain J. A falhng ratio of brain- weight during life
is seen in other animals. Max Weber § tells us that in the lion, at
five weeks, four months, eleven months and lastly when full-grown,
the brain represents the following fractions of the weight of the body :
viz. i/18, 1/80, 1/184 and 1/546. And Kellicott has shewn that in the
dogfish, while certain organs, e.g. pancreas and rectal gland, grow
pari passu with the body, the brain grows in a diminishing ratio,
to be represented (roughly) by a logarithmic curve ||.

In the grown man, Raymond Pearl has shewn brain-weight to
increase with the stature of the indix'idual and to decrease with
his age, both in a straight-hne ratio, or linear regression, as the

* R. C. Robb, Hereditary size-limitation in the rabbit, Journ. Exp. Biol, vi,
1929.

t Cf. H. Vorsteher, Einfiuss d. Gesamtgrosse auf die Zusammensetzung des
Kbrpers; Diss., Leipzig, 1923.

X Oliver Goldsmith argues in his Animated Nature as follows, regarding the un-
likelihood of dwarfs or giants: "Had man been born a dwarf, he could not have
been a reasonable creature; for to that end, he must have a jolt head, and then he
would not have body and blood enough to supply his brain with spirits; or if he
had a small head, proportionable to his body, there would not be brain enough for
conducting life. But it is still worse with giants, etc."

§ Die Sdugethiere, p. 117.

II Amer. Journ. of Anatomy, viii, pp. 319-353, 1908.

188 THE RATE OF GROWTH [ch.

statisticians call it. Thus the following wholly empirical equations
give the required ratios in the case of Swedish males :

Brain-weight (gms.) = 1487-8 — 1-94 x age, or
= 915-06 + 2-86 X stature.

In the two sexes, and in different races, these empirical constants
will be greatly changed*; and Donaldson has further shewn that
correlation between brain-weight and body-weight is much closer
in the rat than in manf.

Weight of

•

entire

Weight of

animal

brain

Ratios

(gm.)

(gm.)

J,

In

r

W

w

w:W y^iv:-^W

M>»»=Fr

Marmoset

335

12-5

1:26

1:20

w = 2-30

Spider monkey

1,845

126

15

M

1-56

Felis minuta

1,234

23-6

52

1-2

2-25

F. domestica

3,300

31

107

2-4

2-36

Leopard

27,700

164

168

12

200

Lion

119,500

219

546

1-3

217

Dik-dik

4,575

37

124

2-7

2-30

Steinbok

8,600

49-5

173

2-9

2-32

Impala

37,900

148-5

255

2-75

211

Wildebeest

212,200

443

479

2-8

201

Zebra

255,000

541

472

2-7

1-98

»>

297,000

555

536

2-8

200

Rhinoceros

765,000

655

1170

3-6

209

Elephant

3,048,000

5,430

560

20

1-74

Whale (Globiocephalus)

1,000,000

2,511

400

20

1-77

Mean 2-23 206

Brandt, a very philosophical anatomist, argued some seventy
years ago that the brain, being essentially a hollow structure, a
surface rather than a mass, ought to be equated with the surface
rather than the mass of the animal. This we may do by taking
the square-root of the brain-weight and the cube-root of the body-
weight; and while the ratios so obtained do not point to equality,
they do tend to constancy, especially if we hmit our comparison to
similar or related animals. Or we may vary the method, and ask
(as Dubois has done) to what power the brain-weight must be raised

* Biometrika, iv, pp. 13-105, 1904.

t H. H. Donaldson, A comparison of the white rat with man, etc., Boas Memorial
Volume, New York, 1906, pp. 5-26.

Ill] OF PARTS OR ORGANS 189

to equal the body-weight; and here again we find the same tendency
towards uniformity*.

The converse to the unequal growth of organs is found in their
unequal loss of weight under starvation. Chossat found, in a
well-known experiment, that a starved pigeon had lost 93 per cent,
of its fat, about 70 per cent, of hver and spleen, 40 per cent, of its
muscles, and only 2 per cent, of brain and nervous tissues f. The
salmon spends many' weeks in the river before spawning, without
taking food. The muscles waste enormously, but the reproductive
bodies continue to grow.

As the internal organs of the body grow at different rates, so that
their ratios one to another alter as time goes on, so is it with those
hnear dimensions whose inconstant ratios constitute the changing
form and proportions of the body. In one of Quetelet's tables
he shews the span of the outstretched arms from year to year, com-
pared with the vertical stature. It happens that height and span
are so nearly co-equal in man that direct comparison means little ;
but the ratio of span to height (Fig. 48) undergoes a significant and
remarkable change. The man grows faster in stretch of arms than
he does in height, and span which was less at birth than stature b}''
about I per cent, exceeds it by about 4 per cent, at the age of
twenty. Quetelet's data are few for later years, but it is clear
enough that span goes on increasing in proportion to stature. How
far this is due to actual growth of the arms and how far to increasing
breadth of the chest is another story, and is not yet ascertained.

* Cf. A. Brandt, Sur le rapport du poids du cerveau a celui du corps chez
diflFerents aniraaux, Bull, de la Soc. Imp. des naturalistes de Moscou, XL, p. 525,
1867; J. Baillanger, De I'etendu de la surface du cerveau, Ann. Med. Psychol.
XVII, p. 1, 1853; Th. van Bischoff, Das Hirngewicht des Menschen, Bonn, 1880
(170 pp.), cf. Biol. Centralhl. i, pp. 531-541, 1881; E. Dubois, On the relation
between the quantity of brain and the size of the body, Proc. K. Akad. Wetensch.,
Amsterdam, xvi, 1913. Also, Th. Ziehen,^ Maszverhaltnisse des Gehirns, in
Bardeleben's Handh. d. Anatomie des Menschen; P. Warneke, Gehirn u. Korper-
gewichtsbestimmungen bei Saugern, Journ. f. Psychol, u. Neurol, xiii, pp. 355-403,
1909; B. Klatt, Studien zum Domestikationsproblem, Bibliotheca genetica, ii,
1921; etc. The case of the heart is somewhat analogous ; see Parrot, Zooi. Ja^r&.
(System.), vii, 1894; Piatt, in Biol. Centralhl. xxxix, p. 406, 1919.

t C. Chossat, Recherches sur I'inanition, Mem. Acad, des Sci., Paris, 1843,
p. 438.

190

THE RATE OF GROWTH

[CH.

The growth-rates of head and body differ still more; for the
height of the head is no more than doubled, but stature is trebled,

Height of the head in man at various ages *
(After Quetelet, p. 207, abbreviated)

Men Women

Stature

Head

Stature

Head

Age

m.

ra.

Ratio

m.

m.

Rat

Birth

0-50

Oil

4-5

0-49

Oil

4-4

1 year

0-70

015

4-5

0-69

015

4-5

2 years

0-79

017

4-6

0-78

0-17

4-5

3 „

0-86

0-18

4-7

0-85

0-18

4-7

5 „

0-99

019

51

0-97

019

51

10 „

1-27

0-21

6-2

1-25

0-20

6-2

20 „

1-51

0-22

70

1-49

0-21

7

25 ,.

1-67

0-23

7-3

1-57

0-22

71

30 „

1-69

0-23

7-4

1-58

0-22

71

40 „

1-69

0-23

7-4

1-58

0-22

71

Fig. 48.

Ratio of stature in man, to span of outstretched arms.
From Quetelet's data.

between infancy and manhood. Diirer studied and illustrated this
remarkable phenomenon, and the difference which accompanies and

* A smooth curve, very similar to this, is given by Karl Pearson for the growth
in "auricular height" of the girl's head, in Biometrika, ni, p. 141, 1904.

Ill] OF PARTS OR ORGANS 191

results from it in the bodily form of the child and the man is easy
to see.

The following table shews the relative sizes of certain parts and
organs of a young trout during its most rapid development; and
so illustrates in a simple way the varying growth-rates in different
parts of the body*. It would not be difficult, from a picture of the
little trout at any one of these stages, to draw its approximate
form at any other by the help of the numerical data here set
forth. In like manner a herring's head and tail grow longer,
the parts betw^een grow relatively less, and the fins change their
places a httle; the same changes take place with their specific
differences in related fishes, and herring, sprat and pilchard
owe their specific characters to their rates of growth or modes of
increment f.

Trout (Salmo fario) ; proportionate growth of various organs
(From Jenkinson's data)

Days

Total

1st

Ventral

2nd

Tail

Breadth

old

length

Eye

Head

dorsal

fin

dorsal

fin

• of tail

40

100

100

100

100

100

100

100

100

63

130

129

148

149

149

108

174

156

77

155

147

189

(204)

(194)

139

258

220

92

173

179

220

(193)

(182)

155

308

272

106

195

193

243

173

165

173

337

288

Sachs studied the same phenomenon in plants, after a method
in use by Stephen Hales a hundred and fifty years before. On the
growing root of a bean ten narrow zones were marked off, starting
from the apex, each zone a millimetre long. After twenty-four
hours' growth (at a given temperature) the whole ten zones had
grown from 10 to 33 mm., but the several zones had grown very
unequally, as shewn in the annexed table J (p. 192):

* Cf. J. W. Jenkinson, Growth, variability "and correlation in young trout, Bio-
metrika, vin, pp. 444-466. 1912.

I Cf. E. Ford, On the transition from larval to adolescent herring, Journ. Mar.
Biol. Assoc. XVI, p. 723; xviii, p. 977, 1930-31. 8o also in larval eels, tail and
body grow at different rates, which rates differ in different species; cf. Johannes
Schmidt, Meddel. Kommlss. Havsiindersok. 1916; L. Bertin, Bull. Zool. France,
1926, p. 327.

I From Sachs's Textbook of Botany, 1882, p. 820.

192

THE RATE OF GROWTH

[CH.

Graded growth of bean-root

Increment

Increment

Zone

mm.

Zone

mm.

Apex

1-5

6th

1-3

2nd

5-8

7th

0-5

3rd

8-2

8th

0-3

4th

3-5

9th

0-2

5th

1-6

10th

01

"... I marked in the same manner as the Vine, young Honeysuckle shoots,
etc. . . . ; and I found in them all a gradual scale of unequal extensions, those parts
extending most which were tenderest," Vegetable Staticks, Exp. cxxiii.

The lengths attained by the successive zones He very nearly
on a smooth curve or gradient; for a certain law, or principle
of continuity, connects and governs the growth-rates along the
growing axis. This curve has its family likeness to those differential

10

2 3 4 5 6 7 8

Zones
Fig. 49. Rate of growth of bean-root, in successive zones
of 1 mm. each, beginning at the tip.

curves which we have already studied, in which rate of growth was
plotted against time, as here it is plotted against successive spatial
intervals of a growing structure ; and its general features are those
of a curve, a skew curve, of error. Had the several growth-rates
been transverse to the axis, instead of being longitudinal and
parallel to it, they would have given us a leaf-shaped structure,
of which our curve would represent the outline on either side; or
again, if growth had been symmetrical about the axis, it might have
given us a turnip-shaped soHd of revolution. There is always an
easy passage from growth to form.

Ill] CONCERNING GRADIENTS 193

A like problem occurs when we deal with rates of growth in
successive natural internodes; and we may then pass from the
actual growth of the internodes to the varying number of leaves
which they successively produce. Where we have whorls of leaves
at each node, as in Equisetum or in many water-weeds, then the
problem is simphfied; and one such case has been studied by
Ra3rmond Pearl*. In Ceratophyllum the mean number of leaves
increases with each successive whorl, but the rate of increase
diminishes from whorl to whorl as we ascend. On the main stem
the rate of change is very slow; but in the small twigs, or tertiary
branches, it becomes rapid, as we see from the following abbreviated
table :

Number of leaves per whorl on the tertiary branches of
Ceratophyllum

Order of whorl ...

1

2

3

4

5

6

Mean no. of leaves
Smoothed no.

6-55
6-5

8-07
8-0

1

9-00
90

9-20
9-5

9-75
9-8

10-00
100

Raymond Pearl gives a logarithmic formula to fit the case; but
the main point is that the numbers form a graded series, and can
be plotted as a simple curve.

In short, a large part of the morphology of the organism depends
on the fact that there is not only an average, or aggregate, rate of
growth common to the whole, but also a gradation of rate from one
part to another, tending towards a specific rate characteristic of each
part or organ. The least change in the ratio, one to another, of
these partial or locahsed rates of growth will soon be manifested
in more and more striking differences of form; and this is as much
as to say that the time-element, which is imphcit in the idea oigrowthy
can never (or very seldom) be wholly neglected in our consideration
of form I .

A flowering spray of Montbretia or lily-of-the-valley exemplifies
a growth-gradient, after a simple fashion of its own. Along the

* On variation and differentiation in Ceratophylly,m, Carnegie Inst. Publications,
No. 58, 1907; see p. 87.

t Herein lies the easy answer to a contention raised by Bergson, an(J to which
he ascribes much importance, that "a mere variation of size is one thing, and
a change of form is another." Thus he considers "a change in the form of leaves"
to constitute "a profound morphological difference" {Creative Evolution, p. 71).

194 THE RATE OF GROWTH [ch.

stalk the growth-rate falls away ; the florets are of descending age,
from flower tO' bud: their graded differences of age lead to an
exquisite gradation. of size and form; the time-interval between
one and another, or the "space-time relation" between them all,
gives a peculiar quality — ^we may call it phase-beauty — to the
whole. A clump of reeds or rushes shews this same phase-beauty,
and so do the waves on a cornfield or on the sea. A jet of water
is not much, but a fountain becomes a beautiful thing, and the
play of many fountains is an enchantment at Versailles.

On the weight-length coefficient, or ponderal itidex

So much for the visible changes of form which accompany
advancing age, and are brought about by a diversity of rates of
growth at successive points or in different directions. But it often
happens that an animal's change of form may be so gradual as to
pass unnoticed, and even careful measurement of such small changes
becomes difficult and uncertain. Sometimes one dimension is easily
determined, but others are hard to measure with the same accuracy.
The length of a fish is easily measured ; but the breadth and depth
of plaice or haddock are vaguer and more uncertain. We may then
make use of that ratio of weight to length which we spoke of in the
last chapter: viz. that W oc L^, or W ^ kL^, or W/L^ = k, where
k, the "ponderal index," is a constant to be determined for each
particular case*.

We speak of this /; as a "constant," with a mean value specific
to each species of animal and dependent on the bodily proportions
or form of that animal; yet inasmuch as the animal is continually
apt to change its bodily proportions during life, k also is continually
subject to change, and is indeed a very delicate index of such

* This relation, and how important it is, were clearly recognised by Herbert
Spencer in his Recent Discussions in Science, etc., 1871. The formula has been

X y/w

often, and often independently, employed: first perhaps in the form — y- x 100,

by R. Livi, L'indice ponderale, o rapporto tra la statura e il peso, Atti Soc. Romamt
Antropologica, v, 1897. Values of k for man and many animals are given by
H. Przibram, in Form und Formel, 1922. On its use as an index to the condition
or habit of body of an individual, see von Rhode, in Abderhalden's Arbeitsmethoden,
IX, 4. The constant k might be called, more strictly, ki, leaving kf, and k,i for
the similar constants to be deri-ved from the breadth and depth of the fish.

Ill

THE PONDERAL INDEX

195

progressive changes : delicate — because our measurements of length
are very accurate on the whole, and weighing is a still more dehcate
method of comparison.

Thus, in the case of plaice, when we deal with mean values for
large numbers and with samples so far "homogeneous" that they
are taken at one place and time, we find that k is by no means
constant, but varies, and varies in an orderly way, with increasing
size of the fish. The phenomenon is unexpectedly complex, much
more so than I was aware of when I first wrote this book. Fig. 50

120 —

.2

o 110

§ 100

^ U

90

' 1

T — :—t r -j— T- 1 I

1 '

_

December

^

^

—

—

~-y

March

^

^'^"^^■"•..•^

—

, , 1

1 1 1 1 1 1 1 1 1

\ ,

1 1 1 1 1

1 1

22

25

40

30 35

Length (cm.)

Fig. 50. Changes in the weight-length coefficient of plaice with
increasing size; from March and December samples.

shews the weight-length coefficient, or ponderal index, in two
large samples, one taken in the month of March, the other in
December. In the latter sample k increases steadily as the plaice
grow from about 25 to 40 cm. long; weight, that is to say, increases
more rapidly than the cube of the length, and it follows that length
itself is increasing less rapidly than some other linear dimension.
In other words, the plaice grow thicker, or bulkier, with length and
age. The other sample, taken in the month of March, is curiously
different; for now k rises to a maximum when the fish are some-
where about 30 cm. long, and then decHnes slowly with further
increase in size of the fish; and k itself is less in March than in
December, the discrepancy being slight in the small fish and great
in the large. , The "point of inflection" at 30 cm. or thereby marks

196 THE RATE "OF GROWTH [ch.

an epoch in the fish's life; it is about the size when sexual maturity
begins, or at least near enough to suggest a connection between the
two phenomena*.

A step towards further investigation would be to determine k for the two
sexes separately, and to see whether or no the point of inflection occurs, as
maturity is known to be reached, at a smaller size in the male. This d'Ancona
has done, not for the plaice but for the shad {Alosa finta). He finds that the
males are the first to reach maturity, first to shew a retardation of the rate
of growth, first to reach a maximal value of th^ ponderal index, and in all
probability the first to diet •

Again we may enquire whether, or how, k varies with the time
of year; and this torrelation leads to a striking result J. For the
ponderal index fluctuates periodically with the seasons, falling
steeply to a minimum in March or April, and rising slowly to an
annual maximum in December (Fig. 51) §. The main and obvious
explanation hes in the process of spawning, the rapid loss of weight
thereby, and the slow subsequent rebuilding* of the reproductive
tissues; whence it follows that, without ever seeing the fish spawn,
and without ever dissecting one to see the state of its reproductive
system, we may by this statistical method ascertain its spawning
season, and determine the beginning and end thereof with con-
siderable accuracy. But all the while a similar fluctuation, of
much less amphtude, is to be found in voung plaice before the
spawning age; whence we learn that the* fluctuation is not only
due to shedding and replacement of spawn, but in part also to
seasonal changes in appetite and general condition.

Returning to our former instance, we now see that the March
and December samples of plaice, which shewed such discrepant
variations of the ponderal index with increasing size, happen to

♦ The carp shews still more striking changes than does the plaice in the weight -
length coefl&cient: in other words, still greater changes in bodily shape with
advancing age and increasing size; cf. P. H. Stnithers, The Champlain Watershed,
Albany, New York, 1930.

■j- U. d'Ancona, II problema dell' accrescimento dei pesci, etc., Mem. R. Acad,
dei Lincei (6), n, pp. 497-540, 1928.

J Cf. Lammel, Ueber periodische Variationen in Organismen, Biol. Centralbl.
xxn, pp. 368-376, 1903.

§ When we restrict ourselves, for simphcity'a sake, to fish of one particular
size, we need not determine the values of k, for changes in weight are obvious
enough; but when we have small numbers and various sizes to deal with, the
determination of k helps very much.

Ill

THE PONDERAL INDEX

197

coincide with the beginning and end of the spawning season; the
fish were full of spawn in December, but spent and lean in March.
The weight-length ratio was, of necessity, higher at the former
season; and the faUing-off in condition, and in bulk, which the
March sample indicates, is more and more pronounced in the larger
and therefore more heavily spawn-laden fish.

JFMAMJJASONDJ
Fig. 51. Periodic annual change in the weight-length ratio of plaice.

Periodi-c relation of weight to length in plaice of 55 cm. long

Average weight

WJL^

W/L^ (smoothed)

decigrams

Jan.

204

1-23

116

Feb.

174

104

108

March

162

0-97

0-99

April

159

0-95

0-97

May

162

0-98

0-98

June

171

103

101

July

169

101

104

August

178

1-07

104

Sept.

173

104

Ml

Oct.

203

1-22

116

Nov.

203

1-22

1-21

Dec.

200

1-20

1-22

Mean

180

1-08

198 THE RATE OF GROWTH [ch.

Plaice caught in a certain area, March 1907 and December 1905.
Variation of k, the weight-length coefficient, with size

March sample

December sample

A

r

Do.

Do.

cm.

gm.

WjL^ smoothed

gm.

W/L^

smoothed

23

113

0-93

24

128

0-93

0-94

— '

—

25

152

0-97

0-96

26

178

0-96

0-98

177

101

27

193

0-98

0-99

209

106

1-06

28

221

101

1-00

241

MO

1-08

29

250

102

101

264

108

109

30

271

1-00

101

294

109

109

31

300

101

100

325

109

110

32

328

100

100

366

112

112

33

354

0-99

0-99

410

114

113

34

384

0-98

0-98

449

M4

115

35

419

0-98

0-98

501

117

117

36

454

0-97

0-97

556

119

117

37

492

0-95

0-96

589

116

118

38

529

0-96

0-96

652

119

119

39

564

0-95

0-95

719

1-21

1-22

40

614

096

0-95

809

1-26

—

41

647

0-94

0-94

—

42

679

0-92

0-93

—

—

—

43

732

0-92

0-93

—

• —

—

H

800

0-94

0-94

—

—

—

4l5

875

0-96

—

—

—

—

These weights and measurements of plaice are taken from the Department of
Agriculture and Fisheries' Plaice-Report, i, pp. 65, 107, 1908; ii, p. 92, 1909.

Japanese goldfish* are exposed to a much wider range of tem-
perature than our plaice are called on to endure ; they hibernate in
winter and feed greedily in the heat of summer. Their weight is
low in winter but rises in early spring, it falls as low as ever at the
height of the spawning season in the month of May; so for one
weight-length fluctuation which the plaice has, the goldfish has a
twofold cycle in the year. The index reaches its second and higher
maximum in August, and falls thereafter till the end of the year.
That it should begin to fall so soon, and fall so quickly, merely means
that late autumn is a time of groipth ; the fish are not losing weight,
but growing longer f.

* Cf. Kichiro Sasaki, Tohoku Soi. Reports (4), i, pp. 239-260, 1926.

t Much has been written on the weight-length index in fishes. See (int. al.)
A. Meek, The growth of flatfish, Northumberland Sea Fisheries Ctee, 1905, p. 58;
W, J. Crozier, Correlations of weight, length, etc, in the weakfish, Cynoscion

hi] the PONDERAL index 199

It is the rule in fishes and other cold-blooded vertebrates that
growth is asymptotic and size indeterminate, while in the warm-
blooded growth comes, sooner or later, to an end. But the
characteristic form is established earlier in the former case, and
changes less, save for the minor fluctuations we have spoken of.
In the higher animals, such as ourselves, the whole course of life
is attended by constant alteration and modification of form; and

Fig. 52. The ponderal index, or weight-length coefficient, in
man. From Quetelet's data.

we may use our weight-length formula, or ponderal index, to illus-
trate (for instance) the changing relation between height and weight
in boyhood, of which we spoke before (Fig. 52).

regalis, Bull. U.S. Bureau of Fisheries, xxxni, pp. 141-147, 1913; Selig Hecht,
Form and growth in fishes, Journ. of Morphology, xxvii, pp. 379-400, 1916;
J. Johnstone (Plaice), Trans. Liverpool Biolog. Soc. xxv, pp. 186-224, 1911;
J. J. Tesch (Eel), Journ. du Conseil, iii, 1927; Frances N. Clark (Sardine), Calif.
Fish. Bulletin, No. 19, 1928 (with full bibliography). For a discussion on statistical
lines, apart from any assumptions such as the "law of the cubes," see G. Buncker,
Korrelation zwischen Lange u. Gewicht, etc., Wissensch. Meerestmtersuch. Helgoland,
XV, pp. 1-26, 1923.

THE RATE OF GROWTH

[CH.

The weight-length coefficient, or ponderal index, k, in young Belgians
(From Quetelefs figures)

(years)

WJL^

Age (years)

WjL^

2-55

10

1-25

1

2-92

11

118

2

2-34

12

1-23

3

2-08

13

1-29

4

1-87

14

1-23

5

1-72

15

1-23

6

1-56

16

1-28

7

1-48

20

1-30

8

1-39

25

1-36

9

1-29

The infant is plump and chubby/ and the ponderal index is at
its highest at a year old. As the boy grows, it is in stature that he
does so most of all; his ponderal index falls continually, till the
growing years are over, and the lad "fills out" and grows stouter
again. During prenatal Ufe the index varied httle, and less than
we might suppose :

Relation between length and weight of the human foetus
(From Scammon's data)

Length

Weight

cm.

gm.

7-7

13

12-3

41

17-3

115

223

239

27-2

405

32-3

750

37-2

1163

42-2

1758

46-9

2389

51-7

3205

2-9
2-2
2-2
2-2

io

2-2
2-3
2-3
2-3
2-3

As a further illustration of the rate of growth, and of unequal
growth in various directions, we have figures for the ox, extending
over the first three years of the animal's life, and giving (1) the
weight of the animal, month by month, (2) the length of the back,
from occiput to tail, and (3) the height to the withers. To these
I have added (4) the ratio of length to height, (5) the weight-length
coefficient, k, and (6) a similar coefficient, or index-number, k' , for

Ill]

THE PONDERAL INDEX

201

the height, of the animal. All these ratios change as time goes on.
The ratio of length to height increases, at first considerably, for the
legs seem disproportionately long at birth in the ox, as in other

Relations between the weight and certain linear dimensions of the ox

{Data from i

Jo^nevin*y

abbreviated)

Length

Age

Weight

of back

Height

lonths

kgm.

m.

m.

L/H

k = WjL^

k' = Wli

37

0-78

0-70

Ml

0-78

1-08

1

55

0-94

0-77

1-22

0-66

1-21

2

86

1-09

0-85

1-28

0-67

1-41

3

121

1-21

0-94

1-28

0-69

1-46

4

150

1-31

0-95

1-38

0-66

1-75

5

179

1-40

104

1-35

0-65

1-60

6

210

1-48

109

1-36

0-64

1-64

7

247

1-52

112

1-36

0-70

1-75

8

267

1-58

115

1-38

0-68

1-79

9

283

1-62

116

1-39

0-66

1-80

10

304

1-65

119

1-39

0-68

1-79

11

328

1-69

1-22

1-39

0-67

1-79

12

351

1-74

1-24

1-40

0-67

1-85

13

375

1-77

1-25

1-41

0-68

1-90

14

391

1-79

1-26

1-41

0-69

1-94

15

406

1-80

1-27

1-42

0-69

1-98

16

418

1-81

1-28

1-42

0-70

2-09

17

424

1-83

1-29

1-42

0-69

1-97

18

424

1-86

1-30

1-43

0-66

1-94

19

428

1-88

1-31

1-44

0-65

1-92

20

438

1-88

1-31

1-44

0-66

1-94

21

448

1-89

1-32

1-43

0-66

1-94

22

464

1-90

1-33

1-43

0-68

1-96

23

481

1-91

1-35

1-42

0-69

1-98

24

501

1-91

1-35

1-42

.0-71

203

25

521

1-92

1-36

1-41

0-74

2-08

26

534

1-92

1-36

1-41

0-75

212

27

547

1-93

1-36

1-41

0-76

2-16

28

555

1-93

1-36

1-41

0-77

219

29

562

1-93

1-36

1-41

0-78

2-22

30

586

1-95

1-38

1-41

0-79

2-22

31

611

1-97

1-40

1-40

0-80

2-21

32

626

1-98

1-42

1-40

0-80

219

33

641

200

1-44

1-39

0-81

216

34

656

201

1-45

1-38

0-81

213

ungulate animals; but this ratio reaches its maximum and falls off
a little during the third year : so indicating that the beast is growing
more in height than length, at a time when growth in both

* Ch. Cornevin, ^^tudes sur la croissanee, Arch, de Physiol, norm, et pathol. (5), iv,
p. 477, 1892. Cf. also R. Gartner, Ueber das W'achstum d. Tiere, Landxvirtsch.
Jahresher. lvii, p. 707, 1922.

202

THE RATE OF GROWTH

[CH.

dimensions is nearly over*. The ratio W/H^ increases steadily,
and at three years old is double what it was at birth. It is the
most variable of the' three ratios; and it so illustrates the some-
what obvious but not unimportant fact that k varies most for the
dimension which varies least, or grows most uniformly; in other
words, that the values of k, as determined at successive epochs for
any one dimension, are a measure of the variability of the other two.
The same ponderal index serves as an index of "build," or
bodily proportion; and its mean values have been determined for
various ages and f6r many races of mankind. Within one and the

ie>u

1 •

r-

•

y

y-

150

y

y

X«

140

my

X'

y

130

^

1

1 1

1 ,

1

1 1

\

60

70

65

Height (inches)

Fig. 53. Ratio of height to weight in man. From Goringe's data.

same race it varies with stature ; for tall men, and boys too, are apt
to be slender and lean, and short ones to be thickset and strong.
And so much does the weight-length ratio change with build or
stature that, in the following table of mean heights and weights of
men between five and six feet high, it will be seen that weight,
instead of varying as the cube of the height, is (within the hmits
shewn) in nearly simple linear relation to it (Fig. 53) f.

* As a matter of fact, the data shew that the animal grows under 7 per cent,
in length, but over 11 per cent, in height, between the twentieth and the thirtieth
month of its age.

t Had the weights varied as the cube of the height, the tallest men should
have weighed close on 200 lb., instead of 160 lb.

Ill] OF WHALES AND ELEPHANTS 203

Ratio of height to weight in man *

No. of

Height

Weight

instances

in.

lb.

WIH

Wjm

59

60-5

125

207

5-62

118

61-5

129

213

5-55-

220

62-5

133

213

• 5-45

285

63-5

136

214

5-30

327

64-5

139

215

519

386

65-5

143

218

5-09

346

66-5

146

2-20

4-97

289

67-5

153

2-27

4-96

220

68-5

151

2-20

4-71

116

69-5

156

224

4-64

58

70-5

160

2-27

4-57

The same index may be used as a measure of the condition, even
of the quaUty, of an animal; three Burmese elephants had the
following heights, weights, and reputations!:

Height

Weight

WjH^

A
B
C

7 ft. lOi in.

8 1
7 5

7,511 lb.

7,216

4,756

1-54
1-36
115

A famous elephant

A good elephant

A weak, poor elephant

But a great African elephant, 10 ft. 10 in. high, weighed 14,640 Ib.J :
whence the weight-height coefficient was no more than 1-15. That is
to say, the African elephant is considerably taller than the Indian,
and the weight-height ratio is correspondingly less.

Lastly, by means of the same index we may judge, to a first rough
approximation, the weight of a large animal such as a whale, where
weighing is out of the question. Sigurd Rusting has given us
many measurements, and many foetal weights, from the Antarctic
whale-fishery: among which, choosing at random, we find that a
certain foetus of the blue whale, or Sibbald's rorqual, measured
4 ft. 6 in. long, and weighed 23 kilos, or say 46 lb. A whale of the
same kind, 45 ft. long, should then weigh 46 x 10^ lb., or about
23 tons; and one of 90 ft., 23 x 2^ tons, or over 180 tons. Again
in seven young unborn whales, measuring from 39 to 54 inches and
weighing from 10 to 23 kilos, the mean value of the index was found

* Data from Sir C. Goringe, The English Convict, H.M. Stationery Office, 1913.
See also J. A, Harris and others. The, Measurement of Man, Minnesota, 1930, p. 41.
t Data from A. J. Milroy, On the management of elephants, Shillong, 1921.
% D. P. Quireng, in Qrcrwth, iii, p. 9, 1939.

204 THE RATE OF GROWTH [ch.

to be 15-2, in gramme-inches. From this we calculate the weight
of the great rorqual, as follows :

1 5*2 X SOO^

At 25 ft., or 300 inches, W = -— — = 4,100,000 g.

lUu

= 4,100 kg.

= 4 tons, nearly.

At 50 ft., TF = 4 X 23 tons = 32 tons.

100 ft. = 32 X 23 tons = 256 tons.

106 tons (the largest known) W = 305 tons, nearly.

The two independent estimates are in close agreement.

Of surface and volume

While the weight-length relation is of especial importance, and
is wellnigh fundamental to the understanding of growth and form
and magnitude, the corresponding relation of surface-area to weight
or volume has in certain cases an interest of its own. At the surface
of an animal heat is lost, evaporation takes place, and oxygen may
be taken in, all in due proportion as near as may be to the bulk
of the animal; and again the bird's wing is a surface, the area of
which must be in due proportion to the size of the bird. In hollow
organs, such as heart or stomach, area is the important thing rather
than weight or mass; and we have seen how the brain, an organ
not obviously but essentially and developmentally hollow, tends to
shew its due proportions when reckoned as a surface in comparison
with the creature's mass.

Surface cannot keep pace with increasing volume in bodies of
similar form; wing-area does not and cannot long keep pace with
the bird's increasing bulk and weight, and this is enough of itself
to set limits to the size of the flying bird. It is the ratio between
square-root-of-surface and cube-root-of-volume which should, in
theory, remain constant; but as a matter of fact this ratio varies
(up to a certain extent) with the circumstances, and in the case of
the bird's wing with varying modes and capabilities of _flight. The
owl, with his silent, effortless flight, capable of short swift spurts
of attack, has the largest spread of wings of all ; the kite outstrips
the other hawks in spread of wing, in soaring, and perhaps in speed.

Ill] OF^SURFACE AND VOLUME 205

Stork and seagull have a great expanse of wing; but other skilled
and speedy fliers have long narrow wings rather than large ones.
The peregrine has less wing-area than the goshawk or the kestrel;
the swift and the swallow have less than the lark.

Mean ratio, VS/^^^W, between wing-area and weight of birds
{From Mouillard's data)

Ratio

Owls

1 1

species

2-2

Hawks

7

Gulls

1

Waders

3

Petrels

2

Plovers

3

Passeres

4

1-3

Ducks

2

1-2

To measure the length of an animal is easy, to weigh it is easier still, but
to estimate its surface-area is another thing. Hence we know but little of
the surface-weight ratios of animals, and what we know is apt to be uncertain
and discrepant. Nevertheless, such data as we possess average down to mean
values which are more uniform than we might expect*.

Mean ratio, VS/VW, in various animals [cm. gm. units)

Ape

11-8

I Sheep (shorn)

8

Man

11

1 Snake

12-5

Dog

10-11

1 Frog

10-6

Cat, horse

10

Birds

10

Rabbit

9-75

1 Tortoise

10

Cow, pig, rat

9

A further note on unequal growth, or heterogony

An organism is so complex a thing, and growth so complex a
phenomenon, that for growth to be so uniform and constant in all
the parts as to keep the whole shape unchanged would indeed
be an unlikely and an unusual circumstance. Rates ''^ary, propor-
tions change, and the whole configuration alters accordingly. In so
humble a creature as a medusoid,. manubrium and disc grow at
different rates, and certain sectors of the disc faster than others,
as when the little Ephyra-lfnya. "develops" into the great Aurdia-
jellyfish. Many fishes grow from youth to age with no visible,

* From Fr. G. Benedict, Oberflachenbestimmung verschiedener Tiergattungen,
Ergebnisae d. Pkysiologiey xxxvi, pp. 300-346, 1934 (with copious bibliography).

206 THE RATE OF GROWTH [ch.

hardly a measurable, change of form*; but the shapes and looks
of man and woman go on changing long after the growing age is
over, even all their lives long. A centipede has its many pairs of
legs alike, to all intents and purposes; they begin alike and grow
uniformly. But a lobster has his great claws a,nd his small, his
lesser legs, his swimmerets and the broad flaps of his tail ; all these
begin ahke, and diverse rates of growth make up the difference
between them. Moreover, we may sometimes watch a single Umb
growing to an unusual size, perhaps in one sex and not in the other,
perhaps on one side and not on the other side of the body: such
are the "horns," or mandibles, of the stag-beetle, only conspicuous
in the male, and the great unsymmetrical claws of the lobster, or
of that extreme case the httle fiddler-crab {Uca pugnax). For such
well-marked cases of differential growth-ratio between one part and
another, JuHan Huxley has introduced the term heterogonyf.

Of the fiddler-crabs some four hundred males were weighed, in
twenty-five graded samples all nearly of a size, and the weights
of the great claw and of the rest of the body recorded separately.
To begin with the great claw was about 8 per cent., and at the end
about 38 per cent., of the total weight of the unmutilated body.
In the female the claw weighs about 8 per cent, of the whole from
beginning to end; and this contrast marks the disproportionate,
or heterogonic, rate of growth in the male. We know nothing
about the actual rate of growth of either body or claw, we cannot
plot either against time; but we know the relative proportions, or
relative rates of growth of the two parts of the animal, and this is
all that matters meanwhile. In Fig. 54, we have set off the successive
weights of the body as abscissae, up to 700 mgm., or about one-third
of its weight in the adult animal; and the ordinates represent the
corresponding weights of the claw. We see that the ratio between
the two magnitudes follows a curve, apparently an exponential
curve; it does in fact (as Huxley has shewn) follow a compound

* Cf, S. Hecht, Form and growth in fishes, Journ. Morphology, xxvii, pp. 379-
400, 1916; F. S. and D. W. Haramett, Proportional length-growth of garfish
(Lepidosteus), Growth, in, pp. 197-209, 1939.

t See Problems of Relative Growth, 1932, and many papers quoted therein.
The term, as Huxley, tells us, had been used by Pezard; but it Jiad been used, in
another sense, by Rolleston long before to mean an alternation of generations,
or production of offspring dissimilar to the parent.

^73 )([M,j^ J

III] OF UNEQUAL GROWTH 207

interest law, which (calhng y and x the weights of the claw and of
the rest of the body) may be expressed by the usual formula for
compound interest,

y = bx^, or log y = \ogb + k log x;

and the coefficients (6 and Jc) work out in the case of the fiddler-crab,
to begin with, at

2/ = 0-0073 xi-«2. .

300

100 200 300 400 500 600 700
Weight of body (mgm.)

Eig. 54. Relative weights of body and claw in the fiddler-
crab { Uca 'pugnax).

But after a certain age, or certain size, th^e coefficients no longer
hold, and new coefficients have to be found. Whether or no, the
formula is mathematical rather than biological; there is a lack of
either biological or physical significance in a growth-rate which
happens to stand, during part of an animaFs fife, at 62 per cent,
compound interest.

Julian Huxley holds, and many hoki with him, that the exponential
or logarithmic formula, or the compound-interest .law, is of general
application to cases of differential growth-rates. I do not find it to
be so : any more than we have found organ, organism or population
to increase by compound interest or geometrical progression, save

208 THE feATE OF GROWTH [ch.

under exceptional circumstances and in transient phase. Undoubtedly
many of Huxley's instances shew increase by compound interest,
during a phase of rapid and unstinted growth; but I find many
others following a simple-interest rather than a compound-interest
law.

Relative weights of claw and body in fiddler-crabs (Uca pugnax).
(Data abbreviated from Huxley, Problems of Relative Growth,
p. 12)

rt. of body

less claw

Wt. of

Ratio

Wt. of

Wt. of

Ratio

(mgm.)

claw

%

body

claw

%

58

5

8-6

618

243

39-3

80

9

11-2

743

319

42-9

109

14

12-8

872

418

47-9

156

25

160

983

461

46-9

200

38

190

1080

537

49-7

238

53

22-3

1166

594

50-9

270

59

21-9

1212

617

50-9

300

78

260

1299

670

51-6

355

105

29-7

1363

699

51-3

420

135

321

1449

773

53-7

470

165

351

1808

1009

55-8

536

196

36-6

2233

1380

61-7

In the common stag-beetle {Lv^anus cervus) we have the following
measurements of mandible and elytron or wing-case: which two
organs make up the bulk of, and may for our purpose be held
as constituting, the "total length" of the beetle. Here a simple
equation meets the case; in other words, the length of elytron or
of mandible plotted against total length gives what is to all intents
and purposes a straight hne, indicating a simple-interest rather
than a compound-interest rate of increase.

Measurements of iS stag-beetles (Lucanus cervus)* (mm.)

Number of specimens

1

4

5

10

5

7

11

5

Length, total (x)

310

38-7

40-5

42-6

450

46-9

49-2

53-6

Length of elytron (y)

250

30-9

31-5

32-6

33-8

351

36-4

39-2

( „ calculated) (y')

26-9

30-8

31-7

32-8

340

350

36-2

38-5

Length of mandible (z)

60

7-8

90

100

11-2

11-9

12-8

14-4

( „ calculated) (z')

5-9

7-7

9-3

101

110

11-7

12-6

14-2

* Data, from Julian Huxley, after W. Bateson and H. H. Brindley, in P.Z.S.
1892, pp. 585-594.

Ill] OF UNEQUAL GROWTH OR HETEROGONY 209

From the observed data we may solve, by the method of least
squares, the simple equations

y = a -{- bx, z= c + dx,

or in other words, find the equations of the straight lines in closest
agreement with the observed data. The solutions are as follows*:

y - 11-02 + 0-512^, and z^- 5-64 + 0-368x,

the two coefficients 0-368 and 0-512 signifying the difference between
the rates *of increase of the two organs. The number of samples is
not very large, and some deviation is to be expected; nevertheless,
the calculated straight lines come close to the observed values.

40

10

20

Fig.

30 40 50

Total length (mm.)

55. Relative growth of body and mandible in reindeer-beetle
{Cydommatus tarandus).

The reindeer-beetle {Cydommatus tarandus), belonging to the
same family, shews much the same thing. The mandible grows in
approximately linear ratio to the body, save that it tends to be at
first a little above, and later on a little below, this Unear ratio
(Fig. 55).

Measurements o/ Cydommatus tarandus f (mm.)

Length of mandible (y)

3-9

10-7

^ 141

19-9

24-0

30-7

34-5

Total length (x)

20-4

331

38-4

47-3

54-2

66-1

74-0

Total length calculated:
x = l7y + U-l

20-3

31-9

37-7

47-5

54*5

65-9

72-4

* As determined for me by Dr A. C. Aitken, F.R.S.

t Data, much abbreviated, from Huxley, after E. Dudieh, Archivf. Naturgesch. (A),
1923.

22-0

48-3

58-0

73-5

891

102-0

1120

420

65-3

74-5

85-5

99-3

112-6

1200

421

65-2

73-7

87-4

971

112-5

121-2

210 THE RATE OF GROWTH [ch.

The facial and cranial parts of a dog's skull tend to grow at
different rates (Fig. 56) ; and changes in the ratio between the two
go a long way to explain the differences in shape between one dog's
skull and another's, between the greyhound's and the pug's. But
using Huxley's own data (after Becher) for the sheepdog, I find the
ratio between the facial and cranial portions of the skull to be, once
again, a simple linear one.

Measurements of skull of sheep-dog (30 specimens) * {mm.)

Mean length of facial region (y)

Mean length of cranial region- (x)

Calculated values for cranial
region: x = 22-7 +0-88y

And now, returning to the fiddler-crab, we find that after the
crab has reached a certain size and the first phase of rapid growth
is over, claw and body grow in simple linear relation to one another,
and the heterogonic or compound-interest formula is no longer
required:

Fiddler-crab (Uca pugnax): ratio of growth-rates ^ in later stages ^
of claw and body (mgm.)

Weight of body less 872 983 1080 1165 1212 1291 1363 1449
claw {x)

Weight of large 418 461 537 594 617 670 699 778

claw {y)

Do., calculated: 413 480 538 590 617 665 708 759

y=0-6a;-110

* Data from A. Becher, in Archivf. Naturgesch. (A), 1923; see Huxley, Problems
of Relative Growth, p. 18, and Biol. Centralbl. loc. cit. Here, and in the previous
case of Cydommatus, the equation has been arrived at in a very simple way. Take
any two values, x^, x^, and the corresponding values, yi, y^. Then let

x-Xj^ ^ y-yi
^-^ Vz-Vx
Of -65-3 v-48-3

e.g.

or

from which a; = 22-7+0-88y.

We may with advantage repeat this process with other values of x and y; and
take the mean of the results so obtained.

112-6-65-3 1020-48-3^

a; -653 _y-48-3
47-3 "■ 53-7 *

TIT]

OF HETEROGONY

211

Once again we find close agreement between the observed and
calculated values, although the observations are somewhat few and
the equation is arrived at in a simple way. We may take it as
proven that the relation between the two growth-rates is essentially
linear.

A compound-interest law of growth occurs, as Malthus knew,
in cases, and at times, of rapid and unrestricted growth. But
unrestricted growth occurs under special conditions and for brief

50 100

Length of cranial portion (mm.)

150

Fig. 56. Relative growth of the cranial and facial portions of the
skull in the sheepdog. Cf. Huxley, p. 18, after Becher,

periods; it is the exception rather than the rule, whether in a
population or in the single organism. In cases of diiferential
growth the compound-interest law manifests itself, for the same
reason, when one of the two growth-rates is rapid and "unre-
stricted," and when the discrepancy between the two growth-rates is
consequently large, for instance in the fiddler-crabs. The compound-
interest law is a very natural mode of growth, but its range is

212

THE RATE OF GROWTH

[CH.

limited. A linear relation, or simple-interest law, seems less likely
to occur; but the fact is, it does occur, and occurs commonly.

On so-called dimorphism

In a well-known paper, Bateson and Brindley shewed that among
a large number of earwigs collected in a particular locality, the
males fell into two groups, characterised by large or by small

DD

Fig. 57. Tail-forceps of earwig. From Martin Burr, after Willi Kuhl.
150

^ 100

mm. 10

Length in mm.

Fig. .58. Variability of length of tail-forceps in a sample of earwigs.

After Bateson and Brindley, P.Z.S. 1892, p. 588.

tail-forceps (Fig. 57), with few instances of intermediate magnitude*.
This distribution into two groups, according to magnitude, is
illustrated in the accompanying diagram (Fig. 58); and the

* W. Bateson and H. H. Brindley, On some cases of variation in secondary
sexual characters [Forficula, Xylotrupa], statistically examined, P.Z.S. 1892,
pp. 585-594. Cf. D. M. Diakonow, On dimorphic variability of Forficula, Joum.
Genet, xv, pp. 201-232, 1925; and Julian Huxley, The bimodal cephalic horn of
Xylotrupa, ibid, xviii, pp. 45-53, 1927.

Ill

OF DIMORPHIC GROWTH

213

phenomenon was described, and has been often quoted, as one
of dimorphism or discontinuous variation. In this diagram the
time-element does not appear; but 'it looks as, though it lay close
behind. For the two size-groups into which the tails of the earwigs
fall look curiously hke two age-groups such as we have already
studied in a fish, where the ages and therefore also the magnitudes of a
random sample form a discontinuous series (Fig. 59). And if, instead
of measuring the whole length of our fish, we had confined ourselves
to particular parts, .such as' head, or tail or fin, we should have
obtained discontinuous curves of distribution for the magnitudes

200

150

100

Ocm. \5 20 25 30

Length in cm.
Fig. 59. Length of body in a random sample of plaice.

of these organs, just as for the whole body of the fish, and just as
for the tails of Bateson's earwigs. The differences, in short, with
which Bateson was dealing were a question of magnitude, and it
was only natural to refer these diverse magnitudes to diversities of
growth; that is to say, it seemed natural to suppose that in this
case of "dimorphism," the tails of the one group of earwigs (which
Bateson called the "high males") had either grown faster, or had
been growing for a longer period of time, than those of the "low
males." If the whole random sample of earwigs were of one and
the same age, the dimorphism would appear to be due to two
alternative values for the mean growth-rate, individual earwigs
varying around one mean or the other. If, on the other hand, the

214

THE RATE OF GROWTH

[CH.

two groups of earwigs were of diiferent ages, or had passed through
one moult more or less, the phenomenon would be simple indeed,
and there would be no more to be said about it*. Diakonow made
the not unimportant observation that in earwigs living in un-
favourable conditions only the short-tailed type tended to appear.
In apparent close analogy with the case of the earwigs, and in
apparent corroboration of their dimorphism being due to age,
Fritz Werner measured large numbers of water-fleas, all apparently
adult, found his measurements falling into groups and so giving
multimodal curves. The several cusps, or modes, he interpreted
without difficulty as indicating diiferences of age, or the number
of moults which the creatures had passed through f (Fig. BO).

9
/i. 60

10 II 12 13 14 15 16 17 18 19 20 21 22 23 24

80 100 120 140 160/1

Length in ^

Fig. 60. Measurements of the dorsal edge in a population of
Chydorus sphaericus, a water-flea.

From Fritz Werner.

An apparently analogous but more difficult case is that of a
certain little beetle, Onthofhagus taurus, which bears two "horns"
on its head, of variable size or prominence. Linnaeus saw in it
a single species, Fabricius saw two; and the question long remained
an open one among the eniomologists. We now know that there
are two "modes," two predominant sizes in a continuous range of

* The number of moults is known to be variable in many species of Orthoptera,
and even occasionally in higher insects; and how the number of moults may be
influenced by hunger, damp or cold is discussed by P. P. Calvert, Proc. Amer.
Pkilos. Soc. Lxviii, p. 246, 1929. On the number of moults in earwigs, see E. B.
Worthington, Entomologist, 1926, and W. K. Weyrauch, Biol. Centralbl. 1929,
pp. 543-5^8.

■f Fritz Werner, Variationsanalytische Untersuchungen an Chydoren, Ztschr. f.
Morphologie u. Oekologie d. Tiere, n, pp. 58-188, 1924.

Ill]

OF DIMORPHIC GROWTH

215

variation*. In the "complete metamorphosis" of a beetle there
is no room for a moult more or less, and the reason for the two
modal sizes remains hidden (Fig. 61).

But new hght has been thrown on the case of the earwigs, which
may help to explain other obscure diversities of shape and size
within the class of insects. At metamorphosis, and even in a simple
moult, the external organs of an insect may often be seen to unfold,
as do, for instance, the wings of a butterfly; they then quickly
harden, in a form and of a size with which ordinary gradual growth

I

200

Head of 0.'tauru8;two
forms of maJe

^N

A.

0*5 1-0

Length of horn (mm.)
Fig. 61. Two forms of the male, in the beetle Ontkophagvs taurus.

has had nothing directly to do. This is a very peculiar phenomenon,
and marks a singular departure from the usual interdependence of
growth and form. When the nymph, or larval earwig, is about to
shed its skin for the last time, the tail-forceps, still soft and tender,
are folded together and wrapped in a sheath; they need to be
distended, or inflated, by a combined pressure of the body-fluid
(or haemolymph) and an intake of respiratory air. If all goes well,

* Rene Paulian, Bull. Soc. Zool. Fr. 1933; also Le polymorpMsme des mMea de
Cddoptires, Paris, 1935, p. 8.

216 THE RATE OF GROWTH [ch.

the forceps expand to their full size; if the .reature be weak or
underfed, inflation is incomplete and tho tail-forceps remain small.
In either case it is an affair of a few critical moments during the
final ecdysis; in ten minutes or less, the chitin has hardened, and
shape and size change no- more. Willi Kuhl, who has given us this
interesting explanation, suggests that the dimorphism observed by
Bateson and by Diakonow is not an essential part of the pheno-
menon; he has found it in one instance, but in other and much larger
samples he has found all gradations, but only a single,- well-marked
unimodal peak*.

The effect of temperature'f

The rates of growth which we have hitherto dealt with are mostly
based on special investigations, conducted under particular local
conditions; for instance, Quetelet's data, so far as we have used
them to illustrate the rate of growth in man, are drawn from his
study of the Belgian people. But apart from that "fortuitous"
individual variation which we have already considered, it is obvious
that the normal rate of growth will be found to vary, in man and
in other animals, just as the average stature varies, in different
locahties and in different "races." This phenomenon is a very
complex one, and is doubtless a resultant of many undefined con-
tributory causes; but we at least gain something in regard to it
when we discover that rate of growth is directly affected by
temperature, and doubtless by other physical conditions. Reaumur
was the first to shew, and the observation was repeated by Bonnet {,
that the rate of growth or development of the chick was dependent
on temperature, being retarded at temperatures below and somewhat

* Willi Kuhl, Die Variabilitat der abdominalen Korperanhange bei Forficula,
Ztsch. Morph. u. Oek. d. Tiere, xii, p. 299, 1924. Cf. Malcolm Burr, Discovery,
1939, pp. 340^345.

t The temperature limitations of life, and to some extent of growth, are sum-
marised for a large number of species by Davenport, Exper. Morphology, cc. viii,
xviii, and by Hans Przibram, Exp. Zoologie, iv, c. v.^

X Reaumur, L'art de faire eclorre et elever en toute saison des oiseaux domestiques,
soil par le moyen de la chaleur du fumee, soil par le moyen de celle du feu ordinaire,
Paris, 1749. He had also studied, a few years before, the effects of heat and cold
on growth-rate and duration of life in caterpillars and chrysalids: Memoires, ii,
p. 1, de la dnree de la vie des crisalides (1736). See also his Observations du
Thermometre, etc., Mem. Acad., Paris, 1735, pp. 345-376.

Ill] THE EFFECT' OF TEMPERATURE 217

accelerated at temperatures above the normal temperature of
incubation, that is to say the temperature of the sitting- hen. In
the case of plants the fact that growth is greatly affected by tem-
perature is a matter of famiUar knowledge; the subject was first
carefully studied by Alphonse De Candolle, and his results and those
of his followers are discussed in the textbooks of botany*.

That temperature is only one of the climatic factors determining growth and
yield is well known to agriculturists; and a method of "multiple correlation"
has been used to analyse the several influences of temperature and of rainfall
at different seasons on the future yield of our own crops f. The same joint
influence can be recognised in the bamboo; for it is said (by Lock) that the
growth-rate of the bamboo in Ceylon is proportional to the humidity of the
atmosphere, and again (by Shibata) that it is proportional to the temperature
in Japan. But BlackmanJ suggests that in Ceylon temperature conditions
are all that can be desired, but moisture is apt to be deficient, while in Japan
there is rain in abundance but the average temperature is somewhat low:
so that in the one country it is the one factor, and in the other country the
other, whose variation is both conspicuous and significant. After all, it is
probably rate of evaporation, the joint result of temperature and humidity,
which is the crux of the matter§. "Climate" is a subtle thing, and includes
a sort of micro-meteorology. A sheltered corner has a climate of its own; one
side of the garden -wall has a different climate to the other; and deep in the
undergrowth of a wood celandine and anemone enjoy a climate many degrees
warmer than what is registered on the screen ||.

Among the mould-fungi each several species has its own optimum tempera-
ture for germination and growth. At this optimum temperature growth is
further accelerated by increase of humidity ; and the further we depart from
the optimum temperature, the narrower becomes the range of humidity within
which growth can proceed^. Entomologists know, in like manner, how over-
abundance of an insect-pest comes, or is apt to come, with a double optimum
of temperature and humidity.

* Cf. {int. at.) H. de Vries, Materiaux pour la connaissance de I'influence de la
temperature sur les plantes, Arch. Neerlandaises, v, pp. 385-401, 1870; C. Linsser,
Periodische Erscheinungen des Pflanzenlebens, Mem. Acad, des Sc, 8t Petersbourg
(7), XI, XII, 1867-69; Koppen, Warme und Pflanzenwachstum, Bull. Soc. Imp.
Nat., Moscou, xliii, pp. 41-110, 1871; H. Hoffmann, Thermisehe Vegetations-
constanten, Ztschr. Oesterr. Ges. f. Meteorologie, xvii, pp. 121-131, 1881; Pheno-
logische Studien, Meteorolog. Ztschr. iii, pp. H3-120, 1886.

t See [int. al.) R. H. Hooker, Journ. Roy. Statist. Soc. 1907, p. 70; Journ. Roy.
Meteor. Soc. 1922, p. 46.

I F. F. Blackman, Ann. Bot. xix, p. 281, 1905.

§ Szava-Kovatz, in Petermann's Mitteilungen, 1927, p. 7.

II Cf. E. J. Salisbury, On the oecological aspects of Meteorology, Q.J.R. Meteorol.
Soc. July 1939.

^ R. G. Tomkins, Proc. R.S. (B), cv, pp. 375^01, 1929.

218

THE RATE OF GROWTH

[CH.

The annexed diagram (Fig. 62), showing growth in length of the
roots of some common plants at various temperatures, is a sufl&cient
illustration of the phenomenon. We see that there is always a
certain temperature at which the rate is a maximum ; while on either
side of the optimum the rate falls off, after the fashion of the normal
curve of error. We see further, from the data given by Sachs and
others, that the optimum is very much the same for all the common
plants of our own chmate. For these it is somewhere about 26° C.

80

70

M^ie 16 20 22 24 26 28 30 32 34 36 38 40°

Temp.

Fig. 62. Relation of rate of growth to temperature in certain

plants. From Sachs's data.

(say 77° F.), or about the temperature of a warm summer's day;
while it is considerably higher, naturally, in such plants as the melon
ot* the maize, which are at home in warmer countries than our own.
The bacteria have, in like manner, their various optima, and some-
times a high one. The tuberculosis-bacillus, as Koch shewed, only
begins to grow at about 28° C, and multipUes most rapidly at
37-38°, the body- temperature of its host.

The setting and ripening of fruit is a phase of growth stiU more
dependent on temperature; hence it is a "dehcate test of climate,"
and a proof of its constancy, that the date-palm grows but bears

Ill] THE EFFECT OF TEMPERATURE 219

no fruit in Judaea, and the vine bears freely at Eshcol, but not in
the hotter country to the south*. Shellfish have their own appro-
priate spawning-temperatures; it needs a warm summer for the
oyster to shed her spat, and Hippopus and Tridacna, the great clams
of the coral-reefs, only do so when the water has reached the high
temperature of 30° C. For brown trout, 6° C. is found to be a
critical temperature, a minimum short of which they do not grow
at all; it follows that in a Highland burn their growth is at a
standstill for fully half the yearf.

That a rise of temperature accelerates growth is but part of the
story, and is not always true. Several insects, experimentally
reared, have been found to diminish in size as the temperature
increased J; and certain flies have been found to be larger in their
winter than their summer broods. The common copepod, Calmius
finmarchicus, has spring, summer and autumn broods, which (at
Plymouth §) are large, middle-sized and small; but the large spring
brood are hatched and reared in the cold "winter" water, and the
small autumn-winter brood in the warmest water of the year. In
the cold waters of Barents Sea Calanus grows larger still; of an
allied genus, a large species lives in the Antarctic, a small one in
the tropics, a middle-sized is common in the temperate oceans.
The large size of many Arctic animals, coelenterates and crustaceans,
is well known; and so is that of many tropical forms, Hke Fungia
among the corals, or the great Tritons and Tridacnas among
molluscs. Another common phenomenon is the increasing number
of males in late summer and autumn, as in the Rotifers and in the
above-mentioned Calani. All these things seem somehow related
to temperature; but other physical conditions enter iilto the case,
for instance the amount of dissolved oxygen in the cold waters, and
the physical chemistry of carbonate of Hme in the warm||.

The vast profusion of life, both great and small, in Arctic seas, the multitude
of individuals and the unusual size to which many species grow, has been
often ascribed to a superabundance of dissolved oxygen, but oxygen alone
would not go far. The nutrient salts, nitrates and phosphates, are the

* Cf. J. W. Gregory, in Geogr. Journ. 1914, and Journ. E. Geogr. Soc. Oct. 1930.

t Cf. C. A. Wingfield, op. cit. supra, p. 176.

X B. P. Uvarow, Trans. Ent. Soc. Lond. lxxix, p. 38, 1931.

§ W. H. Golightly and LI. Lloyd, in Nature, July 22, 1939.

li Cf. B. G. Bogorow and others, in the Journ. M.B.A. xix, 1933-34.

220 THE RATE OF GROWTH [ch.

limiting factor in the growth of that micro-vegetation with which the whole
cycle of life begins. The tropical oceans are often very bare of these salts;
in our own latitudes there is none too much, and the spring-growth tends to
use up the supply. But we have learned from the Discovery Expedition
that these salts are so abundant in the Antarctic that plant-growth is never
checked for stint of them. Along the Chilean coast and in S.W. Africa,
cold Antarctic water wells up from below the warm equatorial current. It
is ill-suited for the growth of corals, which build their reefs in the warmer
waters of the eastern side ; but it teems with nourishment, breeds a plankton-
fauna of the richest kind, which feeds fishes preyed on by innumerable birds,
the guano of which is sent all over the world. Now and then persistent winds
thrust the cold current aside ; a new warm current, el Nino of the Chileans,
upsets the old equilibrium ; the fishes die, the water stinks, the birds starve.
The same thing happens also at Walfisch Bay, where on such rare occasions
dead fish lie piled up high along the shore.

It is curiously characteristic of certain physiological reactions,
growth among them, to be affected not merely by the temperature
of the moment, but also by that to which the organism has been
previously and temporarily exposed. In other words, acclimatisation
to a certain temperature may continue for some time afterwards to
affect all the temperature relations of the body*. That temporary
c<!>ld may, under certain circumstances, cause a subsequent accelera-
tion of growth is made use of in the remarkable process known as
vernalisation. An ingenious man, observing that a winter wheat failed
to flower when sown in spring, argued that exposure to the cold of
winter was necessary for its subsequent rapid growth ; and this he
verified by " chiUing" his seedlings for a month to near freezing-point,
after which they grew quickly, and flowered at the same time as the
spring wheat. The economic advantages are great of so shortening
the growing period of a crop as to protect it from autumn frosts in a
cold chmate or summer drought in a hot one ; much has been done,
especially by Lysenko in Russia, with this end in viewf.

The most diverse physiological processes may be afl'ected by
temperature. A great astronomer at Mount, Wilson, in California,
used some idle hours to watch the "trail-running" ants, which run
all night and all day. Their speed increases so regularly with the
temperature that the time taken to run 30 cm. suffices to tell the

* Cf. Kenneth Mellanby, On temperature coefficients and acclimatisation,
Nature, 3 August 1940.
t Of. {int. al.) V. H. Blackman, in Nature, June 13, 1936.

Ill] THE TEMPERATURE COEFFICIENT 221

temperature to 1° C. ! Of two allied species, one ran nearly half as
fast again as the other, at the same temperature*.

While at low temperatures growth is arrested and at temperatures
unduly high hfe itself becomes impossible, we have now seen that
within the range of more or less congenial temperatures growth
proceeds the faster the higher the temperature. The same is true
of the ordinary reactions of chemistry, and here Van't Hoff and
Arrheniusf have shewn that a definite increase in the velocity of
the reaction follows a definite increase of temperature, according to
an exponential law: such that, for an interval of n degrees the
velocity varies as x", x being called the "temperature coefficient"
for the reaction in question J. The law holds good throughout a
considerable range, but is departed from when we pass beyond
certain normal limits ; moreover, the value of the coefficient is found
to keep to a certain order of magnitude — somewhere about 2 for
a temperature-interval of 10° C. — which means to say that, the
velocity of the reaction is just about doubled, more or less, for a
rise of 10° C.

This law, which has become a fundamental principle of chemical
mechanics, is applicable (with certain qualifications) to the pheno-
mena of vital chemistry, as Van't Hoff himself was the first to declare ;
and it follows that, on much the same fines, one may speak of a
"temperature coefficient" of growth. At the same time we must
remember that there is a very important difference (though we need
not call it a fundamental one) between the purely physical and the

* Harlow Shapley, On the thermokinetics of Dolichoderine ants, Proc. Nat.
Acad. Sci. x, pp. 436-439, 1924.

t Van't HofF and Cohen, Studien zur chemischen Dynamik, 1896; Sv. Arrhenius,
Ztschr. f. phys. Chemie, iv, p. 226.

X For various instances of a temperature coefficient in physiological processes,
see (e.g.) Cohen, Physical Chemistry f or ... Biologists (Enghsh edition), 1903;
Kanitz and Herzog in Zeitschr. f. Elektrochemie, xi, 1905; F. F. Blackman, Ann.
Bot. XIX, p. 281, 1905; K. Peter, Arch.f. Entw. Mech. xx, p. 130, 1905; Arrhenius,
Ergebn. d. Physiol, vii, p. 480, 1908, and Quantitative Laws in Biological Chemistry,
1915; Krogh in Zeitschr. f.allgem. Physiologie,xyi, -pp. 163,178,1914; James Gray,
Proc. E.S. (B), xcv, pp. 6-15, 1923; W. J. Crozier, many papers in Journ. Gen.
Physiol. 1924; J. Belehradek, in Biol. Reviews, v, pp. 1-29, 1930. On the general
subject, see E. Janisch, Temperaturabhangigkeit biologischer Vorgange und ihrer
kurvenmassige Analyse, Pfluger's Archiv, ccix, p. 414, 1925; G. and P. Hertwig,
Regulation von Wachstum . . . durch Umweltsfaktoren, in Hdb. d. normal, u. pathol.
Physiologie, xvi, 1930.

222 THE RATE OF GROWTH [ch.

physiological phenomenon, in that in the former we study (or seek
and profess to study) one thing at a time, while in the living body
we have constantly to do with factors which interact and interfere;
increase in the one case (or change of any kind) tends to be con-
tinuous, in the other case it tends to be brought, or to bring
itself, to arrest. This is the simple meaning of that Law of
Optimum, laid down by Errera and by Sachs as a general principle
of physiology ; namely that every physiological process which varies
(Hke growth itself) with the amount or intensity of some external
influence, does so under such conditions that progressive increase is
followed by progressive decrease; in other words, the function has
its optimum condition, and its curve shews a definite maximu^n.
In the case of temperature, as Jost puts it, it has on the one hand
its accelerating effect, which tends to follow Van't Hoff's law. But
it has also another and a cumulative effect upon the organism:
"Sie schadigt oder sie ermiidet ihn, und je hoher sie steigt desto
rascher macht sie die Schadigung geltend und desto schneller scbxeitet
sie voran*." It is this double effect of temperature on the organism
which gives, or helps to give us our "optim'im" curves, which (like
all other curves of frequency or error) are the expression, not of a
single solitary phenomenon,' but of a more or less complex resultant.
Moreover, as Blackman and others have pointed out, our "optimum"
temperature is ill-defined until we take account also of the duration
of our experiment; for a high temperature may lead to a short but
exhausting spell of rapid growth, while the slower rate manifested
at a lower temperature may be the best in the end. The mile and
the hundred yards are won by different runners; and maximum
rate of worldng,^ and maximum amount of work done, are two very
different things f.

In the case of maize, a certain series of experiments shewed that
the growth in length of the roots varied with the temperature as
follows J:

* On such limiting factors, or counter-reactions, see Putter, Ztschr. f. aUgem.
Physiologic, xvi, pp. 574-627, 1914.

t Cf. L. Errera, UOptimum, 1896 (Recueil d'oeuvres, Physiologie genirale, pp. 338-
368, 1910) ; Sachs, Physiologie d. Pflanzen, 1882, p. 233; PfeflFer, Pflanzenphysiologie,
n, p. 78, 194; and cf, Jost, Ueber die Reactionsgeschwindigkeit ira Organismus,
Biol. CentraWl. xxvi, pp. 225-244, 1906.

t After Koppen, Bull. Soc. Nat. Moscou, XLin, pp. 41-101, 1871.

Ill] THE TEMPERATURE COEFFICIENT 223

Temperature

Growth in 48 hours

°C.

mm.

18-0

11

23-5

10-8

26-6

29-6

23-5

26-5

30-2

64-6

33-5

69-5

36-5

20-7

us

write our

formula in

the form

V(>^)

= x^, or In

laF..._^-

\o0V = nA

Then choosing two values out of the above experimental series
(say the second and the second-last), we have t = 23-5, n = 10,
and F, V = 10-8 and 69-5 respectively.

. ,. - log 69-5 - log 10-8 ,
Accordmgly, — ^ — — = log x,

0-8414 - 0-034
or ^^ = 0-0808,

and therefore the temperature-coefficient

= antilog 0-0808 = 1-204 (for an interval of 1° C).

This first approximation might be much improved by taking account
of all the experimental values, two only of which we have yet made
use of; but even as it is, we see by Fig. 63 that it is in very fair
accordance with the actual results of observation, within those
particular limits of temperature to which the experiment is confined.
For an experiment on Lupinus albus, quoted by Asa Gray*
I have worked out the corresponding coefficient, but a httle more
carefully. Its value I find to be 1-16, or very nearly identical with
that we have just found for the maize; and the correspondence
between the calculated curve and the actual observations is now
a close one.

Miss I. Leitch has made careful observations of the rate of growth of rootlets
of the Pea; and I have attempted a further analysis of her principal resultsf .

* Asa Gray, Botany, p. 387.

t I. Leitch, Some experiments on the influence of temperature on the rate
of growth in Pisum sativum, Ann. Bot. xxx, pp. 25-46, 1916, especially Table III,
p. 45. Cf. Priestley and Pearsall, Growth studies, Ann. Bot. xxxvi, pp. 224-249,
1922.

224

THE RATE OF GROWTH

[CH.

In Fig. 64 are shewn the mean rates of growth (based on about a hundred
experiments) at some thirty-four different temperatures between 0-8° and
29-3°, each experiment lasting rather less than twenty-four hours. Working
out the mean temperature coefficient for a great many combinations of these
values, I obtain a value of 1-092 per C.°, or 2-41 for an interval of 10°, and
a mean value for the whole series shewing a rate of growth of just about
1 mm. per hour at a temperature of 20°. My curve in Fig. 64 is drawn from
these determinations; and it will be seen that, while it is by no means exact
at the lower temperatures, and will fail us altogether at very high tem-
peratures, yet it serves as a satisfactory guide to the relations between rate
and temperature within the ordinary limits of healthy growth. Miss Leitch

18 20 22 24 26 28 30 32 34°C
Fig. 63. Relation of rate of growth to temperature in maize. Observed
values (after Koppen), and calculated curve.

holds that the curve is not a Van't Hoff curve ; and this, in strict accuracy,
we need not dispute. But the phenomenon seems to me to be one into which
the Van't Hoff ratio enters largely, though doubtless combine'd with other
factors which we cannot determine or eliminate.

While the above results conform fairly well to the law of the
temperature-coefficient, it is evident that the imbibition of water
plays so large a part in the process of elongation of the root or
stem that the phenomenon is as much or more a physical than a
chemical one: and on this account, as Blackman has remarked, the
data commonly given for the rate of growth in plants are apt to
be irregular, and sometimes misleading*. We have abundant

* F. F. Blackman, Presidential Address in Botany, Brit. Assoc. Dublin, 1908.

Ill]

OF TEMPERATURE COEFFICIENTS

225

illustrations, however, among animals, in which we may study the
temperature-coefficient under circumstances where, though the
phenomenon is always complicated, true metabohc growth or
chemical combination plays a larger role. Thus Mile. Maltaux and
Professor Massart* have studied the rate of division in a certain
flagellate, Chilomonas paramoecium, and found the process to take
20

/•

—

/

.7

■~

7

/

/

/

_

/

V

-

.

/

/

—

/

'/.

/

/

1 —

•

/

/

^

—

^i

^^

•

^

*

-

•

^

y

-^

•
*

""

per
hour

20

1-8

1-6

1-4

1-2

1-0

•8

•6

•4

•2

4° 8° 1 2° 1 6° 20° 24° ^ 28° 32° ^

Fig, 64. Relation of rate of growth to temperature in rootlets of
pea. From Miss I. Leitch's data.

29 minutes at 15° C, 12 at 25°, and only 5 minutes at 35° C. These
velocities are in the ratio of 1 : 24: 5-76, which ratio corresponds
precisely to a temperature-coefficient of 24 for each rise of 10°, or
about 1-092 for each degree centigrade, precisely the 'Same as we
have found for the growth of the pea.

By means of this principle we may sometimes throw hght on
apparently compUcated experiments. For instance. Fig. 65 is an

* Rec. de VInst. Bot. de Brzizelles, vi, 1906.

226

THE RATE OF GROWTH

[CH.

illustration, which has been often copied, of 0. Hertwig's work on
the effect of temperature on the rate of development of the tadpole*.

'1 ~fZi

Z ^4^

1 1 1

/ / 1

~/ ~X X-

T_ J- L

«'

/ / 1

1 I 1

1 ill

01 «^ ....

- t 1 14^

"^

- 4 -T ttU-

. I 2^ riii

. t^'^ rtlf

'^

v^ IJ LE

- z^^ Tti-l^t

M

R 1 J^iItX-

»^

VJV ptj

^ ^ t Qj 3

-l

f- /- ziy i

f~

/^^'^ z it-,/

" JJ-

-i^'^-^Z Jl J

\ JV .

t- z ^^zt'^

\ .,o^j> y^ L/.

%^^yjLt

£^^^ V

^ /

^^^-""^^^^^^ .-^

>^ ""^^

'^'^^ ^c^^ -^^ ^^^

^^^^^^^^^ --'

-■^

^ — " ^ — ::h^ "^ "^

j TemneXcLl

uie Ceatign.a.d.e

«*• «• 2^" «/•

/*• //" It,' I-.' IV li" If II' lof 9" a" ^ ef s'

Fig. 65. Diagram shewing time taken (in days), at various temperatures {° C),
to reach certain stages of development in the frog: viz. I, gastrula; II,
medullary plate; III, closure of medullary folds; IV, tail- bud; V, tail and
gills; VI, tail-fin; VII, operculum beginning; VIII, do. closing; IX, first
appearance of hind-legs. From Jenkinson, after 0. Hertwig, 1898.

* 0. Hertwig, Einfluss der Temperatur auf die Entwicklung von Rana fusca
und R. esculenta, Arch. f. mikrosk. Anat. li, p. 319, 1898. Cf. also K. Bialaszewicz,
Beitrage z. Kenntniss d. Wachsthumsvorgange bei Amphibienembryonen, Ball.
Acad. Sci. de Cracovie, p. 783, 1908; Abstr. in Arch. f. Entwicklungsmech. xxviii,
p. 160, 1909: from which Ernst Cohen determined the value of Q^q (Vortrdge iib.
physikal. C hemic f. Arzte, 1901; English edit. 1903).

Ill]

OF TEMPERATURE COEFFICIENTS

227

From inspection of this diagram, we see that the time taken to
attain certain stages of development (denoted by the numbers
III-VII) was as follows, at 20° and at 10° C, respectively.

At 20° C.

At 10° C.

Stage III

2-0

6-5 days

„ IV

2-7

8-1 „

„ V

30

10-7 „

;, VI

4-0

13-5 „

„ VII

50

16-8 ,.

Total

16-7

55-6

25'C. 20° 15° 10° 5'

Fig. 66. Calculated values, corresponding to preceding figure.

That is to say, the time taken to produce a given result at 10°
was (on the average) somewhere about 55-6/16-7, or 3-33, times as
long as was required at 20° C.

We may then put our equation in the simple form,

x^^ = 3-33.
Or, 10 log X = log 3-33 = 0-52244.

Therefore log x = 0-05224,

and X = 1-128.

228 THE RATE OF GROWTH [ch.

That is to say, between the intervals of 10° and 20° C, if it take
m days, at a certain given temperature, for a certain stage of
development to be attained, it will take m x 1-128" days, when the
temperature is n degrees less, for the same stage to be arrived at.

Fig. 66 is calculated throughout from this value; and it will be
found extremely concordant with the original diagram, as regards
all the stages of development and the whole range of temperatures
shewn; in spite of the fact that the coefficient on which it is based
was derived by an easy method from a very few points on the
original curves. In hke manner, the following table shews the
"incubation period" for trout-eggs, or interval between fertihsation
and hatching, at different temperatures * :

Incubation-period of trout-eggs

Temperature

Days' interval

°C.

before hatching

2-8

165

3-6

135

3-9

121

4-5

109

50

103

5-7

96

6-3

89

6-6

81

7-3

73

8-0

65

90

56

10-0

47

111

38

12-2

32

Choosing at random a pair of observations, viz. at 3-6° and 10°,
and proceeding as before, we have

10° - 3-6° = 64°.

Then (64) = '-g,

or 6-4 X log X = log 135 — log 47

= 2-1303 - 1-6721 = 0-4582
and log X = 0-4582 - 6-4 = 0-0716,

X = 1-179.

* Data from James Gray, The growth of fish, Journ. Exper. Biology, vi, p. 126,
1928.

Ill] OF TEMPERATURE COEFFICIENTS 229

Using three other pairs of observations, we have the following
concordant results :

At 12-2° and 2-8^, x - M91

10-0° 3-6° M79

9-0° 5-7°; M78

8-0° 5-0° M65

Mean M8

A very curious point is that (as Gray tells us) the young fish which
have hatched slowly at a low temperature are bigger than those
whose growth has been hastened by warmth.

Again, plaice-eggs were found to hatch and grow to a certain
length (4-6 mm.), as follows*:

Temperature (° C.) Days

41 230

6-1 181

8-0 13-3

10-1 10-3

120 8-3

From these we obtain, as before, the following constants:

At 12° and 8°, x = M3

12° 4-1° M4

10-1° 6-1° M5

8-0° 4-1° M5

Mean M4

The value of x is much the same for the one fish as for the other.

Karl Peter t, experimenting on echinoderm eggs, and making use
also of Richard Hert wig's experiments on young tadpoles, gives the
temperature-coefficients for intervals of 10° C. (commonly written
Qio) as follows, to which I have added the corresponding values
forg,:

Sphaerechinus Qiq =2-15 Qi^ 1-08
Echinus ?'13 1-08

Rana 2-86 Ml

* Data from A. C. Johansen and A. Krogh, Influence of temperature, etc.,
Publ. de Circonstance, No. 68, 1914. The function is here said to be a linear one —
which would have been an anomalous and unlikely thing".

t Der Grad der Beschleunigung tierischer Entwicklung durch erhohte Tem-
peratur, Arch. f. Entw. Mech. xx, p. 130, 1905. More recently Bialaszewicz has
determined the coefficient for the rate of segmentation in Rana as being 2-4 per 10° C.

230 THE RATE OF GROWTH fcH.

These values are not only concordant, but are of the same order
of magnitude as the temperature-coefficient in ordinary chemical
reactions. Peter has also discovered the interesting fact that the
temperature-coefficient alters with age, usually but not always
decreasing as time goes on*:

Sphaerechinus Segmentation Q^q =2-29 Q^ = 1-09
Later stages 2-03 1-07

Echinus Segmentation 2-30 1-09

Later stages 2-08 1-08

Rana Segmentation 2-23 1-08

Later stages 3-34 1-13

Furthermore, the temperature-coefficient varies with the tem-
perature itself, falhng as the temperature rises — a rule which Van't
Hoff shewed to hold in ordinary chemical operations. Thus in Rana
the temperature-coefficient (Qiq) at low temperatures may be as
high as 5-6; which is just another way of saying that at low
temperatures development is exceptionally retarded.

As the several stages of development are accelerated by warmth,
so is the duration of each and all, and of life itself, proportionately
curtailed. The span of life itself may have its temperature-
coefficient — in so far as Life is a chemical process, and Death a
chemical result. In hot climates puberty comes early, and old age
(at least in women) follows soon; fishes grow faster and spawn
earlier in the Mediterranean than in the North Sea. Jacques Loeb f
found (in complete agreement with the general case) that the larval
stages of a fly are abbreviated by rise of temperature; that the
mean duration of life at various temperatures can be expressed by
a temperature-coefficient of the usual order of magnitude ; that this
coefficient tends, as usual, to fall as the temperature rises; and
lastly — what is not a little curious — ^that the coefficient is very much
the same, in fact all but identical, for the larva, pupa and imago of
the fly.

* The diflferences are, after all, of small order of magnitude, as is all the better
seen when we reduce the ten-degree to one-degree coefficients.

t J- Loeb and Northrop, On the influence of food and temperature upon the
duration of life, Journ. Biol Chemistry, xxxii, pp. 103-121, 1917.

Ill

OF TEMPERATURE COEFFICIENTS

231

Temperature-coefficients (Q^q) of Drosophila

Larva

Pupa

Imago

15-20° C.

115

117

118

20-25° C.

106

1-08

107

And Japanese students, studying a little fresh- water crustacean, have
carried the experiment much beyond the range of Van't Hoff 's law,
and have found length of hfe to rise rapidly to a maximum at about
13-14° C, and to fall slowly, in a skew curve, thereafter* (Fig. 67).

20 30 40

Temperature, °C.
Fig. 67. Length of life, at various temperatures, in a water-flea.

If we now summarise the various temperature-coefficients (Q^)
which we have happened to consider, we are struck by their
remarkably close agreement:

Yeast Qi = M3

Lupin

M6

Maize

1-20

Pea

1-09

Echinoids

1-08

Drosophila (mean)"

M2

Frog, segmentation

1-08

„ tadpole

M3

Mean

M2

* A. Terao and T. Tabaka, Duration of life in a water-flea, Moina sp.; Joum.
Imp. Fisheries Inst., Tokyo, xxv, No. 3, March 1930.

232 THE RATE OF GROWTH [ch.

The constancy of these results might tempt us to look on the
phenomenon as a simple one, though we well know it to be highly
complex. But we had better rest content to see, as Arrhenius saw
in the beginning, a general resemblance rather than an identity
between the temperature-coefficients in physico-chemical and
biological processes*.

It was seen from the first that to extend Van't Hoff's law from physical
chemistry to physiology was a bold assumption, to all appearance largely
justified, but always subject to severe and cautious limitations. If it seemed
to simplify certain organic phenomena, further study soon shewed how far
from simple these phenomena were. Living matter is always heterogeneous,
and from one phase to another its reactions change; the temperature-
coefficient varies likewise, and indicates at the best a summation, or integration,
of phenomena. Nevertheless, attempts have been made to go a little further
towards a physical explanation of the physiological coefficient. Van't Hoff
suggested a viscosity-correction for the temperature -coefficient even of an
ordinary chemical reaction; the viscosity of protoplasm varies in a marked
degree, inversely with the temperature, and the viscosity-factor goes, perhaps,
a long way to account for the aberrations of the temperature-coefficient. It
has even been suggested (by Belehradekf) that the temperature- coefficients
of the biologist are merely those of protoplasmic viscosity. For instance, the
temperature-coefficients of mitotic cell-division have been shewn to alter
from one phase to another of the mitotic process, being much greater at the
start than at the end| ; and so, precisely, has it been shewn that protoplasmic
viscosity is high at the beginning and low at the end of the mitotic process §.

On seasonal growth

There is abundant evidence in certain fishes, such as plaice and
haddock, that the ascending curve of growth is subject to seasonal
fluctuations or interruptions, the rate during the winter months
bejng always slower than in the months of summer. Thus the
Newfoundland cod have their maximum growth-rate in June, and
in January-February they cease to grow; it is as though we super-
imposed a periodic annual sine-curve upon the continuous curve of
growth. Furthermore, as growth itself grows less and less from
year to year, so will the difference between the summer and the

♦ Cf. L. V. Heilbronn, Science, lxii, p. 268, 1925.
t J. Belehradek, in Biol. Reviews, v, pp. 30-58, 1930.

X Cf. E. Faure-Fremiet, La cinetique du developpement, 1925; also B. Ephrussi,
C.R. cLxxxii, p. 810, 1926.

§ See {int. al.) L. V. Heilbronn, The Colloid Chemistry of Protoplasm, 1928.

Ill]

OF SEASONAL GROWTH

233

winter rates grow less and less. The fluctuation in rate represents
a vibration which is gradually dying out; the amplitude of the
sine-curve diminishes till it disappears; in short our phenomenon
is simply expressed by what is known as a "damped sine-curve*."

Growth in height of German military cadets, in half-yearly periods

Increment (em.)

Height (cm.

)

f

A

■>

Number
observed

,

^

Winter
^-year

Summer
^-year

Age

October

April

October

Year

12

11-12

139-4

141-0

143-3

1-6

2-3

3-9

80

12-13

143-0

144-5

1474

1-5

2-9

4-4

146

13-14

147-5

149-5

152-5

2-0

3-0

5-0

162

14-1.-)

152-2

155-0

158-5

2-8

3-5

6-3

162

15-16

158-5

160-8

163-8

2-3

30

5-3

150

16-17

163-5

165-4

167-7

1-9

2-3

4-2

82

17-18

167-7

168-9

170-4

1-2

1-5

2-7

22

18-19

169-8

170-6

171-5

0-8

0-9

1-7

6

19-20

170-7

171-1

171-5
Mean

0-4
1-6

0-4
2-2

0-8

cm. 4

years

Fig. 68. Half-yearly increments of growth, in cadets of various ages.
From Daffner's data.

The same thing occurs in man, though neither in his case nor in
that of the fish have we sufficient data for its complete illustration.
We can demonstrate the fact, however, by help of certain measure-
ments of the height of German cadets, measured at half-yearly
intervals t- In the accompanying diagram (Fig. 68) the half-yearly
increments are set forth from the above table, and it will be seen

* The scales, on the other hand, make most of their growth during the int/er-
mediate seasons: and with this peculiarity, that a few broad zones are added to
the scale in spring, and a larger number of narrow circuli in autumn : see Contrib.
to Canadian Biology, iv, pp. 289-305, 1929; Ben Dawes, Growth... in plaice,
Journ. M.B.A. xvii, pp. 103-174, 1930.

t From Daffner, Da^ Wachstum des Menschen^ p. 329, 1902.

234

THE RATE OF GROWTH

CH.

that they form two even and entirely separate series. Danish school-
boys show just the same periodicity of growth in stature.

The seasonal effect on visible growth-rate is much alike in fishes
and in man, in spite of the fact that the bodily temperature of the one
varies with the milieu externe and that of the other keeps constant
to within a fraction of a degree.

While temperature is the dominant cause, it is not the only cause
of seasonal fluctuations of growth; for alternate scarcity and
abundance of food is often, as in herbivorous animals, the ostensible

18

10

a
o

S

/ •^A

-1 1 1 J 1 1 X

/ / M

/ \

/ / '^

\ I \

/ /

V / \

/ /
/ /

\\ / \

y /

\\ / Steers\

/

\

/

\ \ / \

v.^

y

\ \ / \

1

1 1 1 1

\ / HeiVfers

\
\
1 1 I.I t 1 .IN

JUN AUG OCT DEC FEB APR dUN AUG OCT DEC FEB APR

15 20 25 30 35

Age in months

Fig. 69. Seasonal growth of S. African cattle: Sussex half-breeds.
After Schiitte.

reason. Before turnips came into cultivation in the eighteenth
century our own cattle starved for half the year and grew fat the
other, and in many countries the same thing happens still. In
South Africa the rainy season lasts from November to February;
by January the grass is plentiful, by June or July the veldt is
parched until rain comes again. Cattle fatten from January to
March or April; from July to October they put on Httle weight,
or lose weight rather than put it on*.

* Cf. D. J. Schiitte, in Onderstepoort Journal, Oct. 1935.

J

Ill] OF THE GROWTH OF TREES 2^5

The growth of trees

Some sixty years ago Sir Robert Christison, a learned and versatile
Edinburgh professor, was the first to study the "exact measurement"
of the girth of trees*; and his way of putting a girdle round the
tree, and fitting a recording device to the girdle, is copied in the
" dendrographs " t used in forestry today. The Edinburgh beeches
begin to enlarge their trunks in late May or June, when in full leaf,
and cease growing some three months later;' the buds sprout and
the leaves begin their work before the cambium wakens to activity.
The beech-trees in Maryland do likewise, save that the dates are
a little earlier in the year; and walnut-trees on high ground in
Arizona shew a like short season of growth, differing somewhat in
date or "phase," just as it did in Edinburgh, from one year to
another.

Deciduous trees stop growing after the fall of the leaf, but ever-
greens grow all the year round, more or less. This broad fact is
illustrated in the following table, which happens to relate to the

Mean monthly increase in girth of trees at San Jorge, Uruguay : from
C. E. HalVs data. Values given in ^percentages of total annual
increment J

Jan.

Feb.

Mar.

Apr.

May

June

July

Aug.

Sept.

Oct.

Nov. Dec.

Evergreens

91

8-8

8-6

8-9

7-7

5-4

4-3

6-0

9-1

Ill

10-8 10-2

Deciduous trees

20-3

14-6

9-0

2-3

0-8

0-3

0-7

13

3-5

9-9

16-7 210

southern hemisphere, and to the climate of Uruguay. The measure-
ments taken were those of the girth of the tree, in mm., at three
feet from the ground. The evergreens included Pinus, Eucalyptus

* Sir R. Christison, On the exact measurement of trees. Trans. Edinb. Botan. Soc.
XIV, pp. 164-172, 1882. Cf. also Duhamel du Monceau, Des semis, et plantation
des arbres, Paris, 1750. On the general subject see {int. al.) Pfeffer's Physiology
of Plants, II, Oxford, 1906; A. Maliock, Growth of trees, Proc. R.S. (B), xc,
pp. 186-191, 1919. Maliock used an exceedingly delicate optical method, in
which interference-bands, produced by two contiguous glass plates, shew a visible
displacement on the slightest angular movement of the plates, even of the order
of a millionth of an inch.

t W. S. Glock, A. E. Douglass and G. A. Pearson, Principles ... of tree-ring
analysis, Carnegie Inst. Washington, No. 486, 1937; D. T. MacDougal, Tree Growth,
Leiden, 1938, 240 pp.

% Trans. Edinb. Botan. Soc. xviii, p. 456, 1891.

236

THE RATE OF GROWTH

[CH.

and Acacia; the deciduous trees included Quercus, Populus, Robinia
and Melia. The result (Fig. 70) is much as we might expect.
The deciduous trees cease to grow in winter-time, and during
all the months when the trees are bare; during the warm season
the monthly values are regularly graded, approximately in a sine-
curve, with a clear maximum (in the southern hemisphere) about
the month of December. In the evergreens the amplitude of the

Fig. 70. Periodic annual fluctuation in rate of growth of trees in
the southern hemisphere. From C. E. Hall's data.

annual wave is much less ; there is a notable amount of growth all
the year round, and while there is a marked diminution in rate
during the coldest months, there is a tendency towards equality
over a considerable part of the warmer season. In short, the
evergreens, at least in this case, do not grow the faster as the
temperature continues to rise; and it seems probable that some of
them, especially the pines, are definitely retarded in their growth,
either by a temperature above their optimum or by a deficiency of
moisture, during the hottest season of the year.

Fig. 71 shews how a cypress never ceased to grow, but had alternate

Ill

OF THE GROWTH OF TREES

237

spells of quicker and slower growth, according to conditions of
which we are not informed. Another figure (Fig. 72) illustrates the
growth in three successive seasons of the Calif ornian redwood, a near
ally of the most gigantic of trees. Evergreen though the redwood
is, its growth has periods of abeyance; there is a second minimum
about midsummer, and the chief maximum of the year may be that
before or after this.

Fig. 71. Growth of cypress (C. macrocarpa), shewing seasonal periodicity.
From MacDougal's data: smoothed curve.

1931 1932 1933

Fig. 72. Fortnightly increase of girth in Californian redwood (Sequoia
sempervirens), shewing seasonal periodicity. After MacDougal.

In warm countries tree-growth is apt to shew a double maximum,
for the cold of winter and the drought of summer are equally
antagonistic tQ it. Trees grow slower — and grow fewer — the farther
north we go, till only a few birches and willows remain, -stunted and
old; it is nearly a hundred years ago since Auguste Bravais*
shewed a steadily decreasing growth-rate in the forests between
50° and 70° N.

* Recherches sur la croissance du pin silvestre dans le nord de I'Europe, Mdm.
couronnees de VAcad. R. de Belgique, xv, 64 pp., 1840-41.

238

THE RATE OF GROWTH

[CH.

The delicate measuring apparatus now used shews sundry minor
but beautiful phenomena. A daily periodicity of growth is a
common thing* (Fig. 73). In the tree-cactuses the trunk expands
by day and shrinks again after nightfall ; for the stomata close in sun-
light, and transpiration is checked until the sun goes down. But
it is more usual for the trunk to shrink from sunrise until evening
and to swell from sunset until dawn ; for by dayhght the leaves lose

70° F

Black Poplar

feet above ground, 59 inches

July 24

26

Fig. 73. Growth of black poplar, shewing daily periodicity.
After A. Mallock.

water faster, and in the dark they lose it slower, than the roots
replace it. The rapid midday loss of water even at the top of a
tail Sequoia is quickly followed by a measurable constriction of the
trunk fifty or even a hundred yards below f.

* The diurnal periodicity is beautifully shewn in the case of the hop by Johannes
Schmidt, C.R. du Laboratoire Carlsberg, x, pp. 235-248, Copenhagen, 1913.

•f This rapid movement is accounted for by Dixon and Joly's "cohesion-theory"
of the ascent of sap. The leaves shew innumerable minute menisci, or cup-shaped
water-surfaces, in their intercellular air-spaces. As water evaporates from these
the little cups deepen, capillarity increases its pull, and suffices to put in motion
the strands or columns of water which run continuously through the vessels of
wood, and withstand rupture even under a pull of 100-200 atmospheres. See
{int. al.) H. H. Dixon and J. Joly, On the ascent of sap, Phil. Trans. (B), clxxxvi,
p. 563, 1895; also Dixon's Transpiration and the Ascent of Sap, 8vo, London, 1914.

Ill] OF THE SECULAR GROWTH OF TREES 239

In the case of trees, the seasonal periodicity of growth and the
direct influence of weather are both so well marked that we are
entitled to make use of the phenomenon in a converse way, and to
draw deductions (as Leonardo da Vinci did*) as to climate during
past years from the varying rates of growth which the tree has
recorded for us by the thickness of its annual rings. Mr A. E.
Douglass, of the University .of California, has made a careful study
of this question, and I received from him (through Professor H. H.
Turner) some measurements of the average width of the annual
rings in Californian redwood, five hundred yfears old, in which trees
the rings are very clearly shewn. For the first hundred years the
mean of two trees was used, for the next four hundred years the
mean of five; and the means of these (and sometimes of larger
numbers) were found to be very concordant. A correction was
appHed by drawing a nearly straight fine through the curve for the
whole period, which line was assumed to represent the slowly
diminishing mean width of annual ring accompanying the increasing
size, or age, of the tree; and the actual growth as measured was
equated with this diminishing mean. The figures used give, then,
the ratio of the actual growth in each year to the mean growth
of the tree at that epoch.

It was at once manifest that the growth-rate so determined
shewed a tendency to fluctuate in a long period of between 100 and
200 years. I then smoothed the yearly values in groups of 100
(by Gauss's method of "moving averages"),, so that each number
thus found represented the mean annual increase during a century:
that is to say, the value ascribed to the year 1500 represented the
average annual growth during the whole period between 1450 and
1550, and so on. These values, so simply obtained, give us a curve
of beautiful and surprising smoothness, from which we draw the
direct conclusion that the climate of Arizona, during the last five
hundred years, has fluctuated with a regular periodicity of almost
precisely 150 years. I have drawn, more recently, and also from
Mr Douglass's data, a similar curve for a group of pine trees in
Calaveras County |. These trees are about 300 years old, and the

* Cf. J. Playfair McMurrich, Leonardo da Vinci, 1930, p. 247.
t When this was first written I had not seen Mr Douglass's paper On a method
of estimating rainfall by the growth of trees. Bull. Amer. Geograph. Soc. XLVI,

240

THE RATE OF GROWTH

[CH.

data are reduced, as before, to moving averages of 100 years, but
without further correction. The agreement between the growth-
rate of these pines and that of the great Sequoias during the same
period is very remarkable (Fig. 74).

We should be left in doubt, so far as these observations go,
whether the essential factor be a fluctuation of temperature or an
alternation of drought and humidity; but the character of the
Arizona chmate, and the known facts of recent years, encourage
the behef that the latter is the more direct and more important
factor. In a New England forest many trees of many kinds were
studied after a hurricane; they shewed on the whole no correlation^

1440

CQ

Fig. 74. Long-period fluctuation in growth of Arizona redwood (Sequoia), from
A.D. 1390 to 1910; and of yellow pine from Calaveras County, from a.d. 1620
to 1920. (Smoothed in 100-year periods.)

between growth-rate and temperature, with the remarkable exception
(in the conifers) of a clear correlation with the temperature of
March and April, a month or two before the season's growth began.
In a cold spring the melting snows and early rains ran off into the
rivers, in a warm and early one they sank into the soil * ; in other
words, humidity was still the controlling factor. An ancient oak'
tree in Tunis is said to have recorded fifty years of abundant rain.

pp. 321-335, 1914; nor, of course, his great work on Climatic cycles and tree-
growth, Carnegie Inst. Publications, 1919, 1928, 1936. Mr Douglass does not
fail to notice the long period here described, but he is more interested in the
sunspot-cycle and other shorter cycles known to meteorologists. See also (int. al.)
E. Huntingdon, The fluctuating climate of North America, Geograph. Journ.
Oct. 1912; and Otto Pettersson, Climatic variation in historic and prehistoric
time, Svens^M, Hydrografisk-Biolog. Skrifter, v, 1914.

♦ C. J. Lyon, Amer. Assoc. Rep. 1939; Nature, Apr. 13, 1940, p. 595.

Ill] OF THE INFLUENCE OF LIGHT 241

with short intervals of drought, during the eighteenth century ; then,
after 1790, longer droughts and shorter spells of rainy seasons*.

It has been often remarked that our common European trees,
such as the elm or the cherry, have larger leaves the farther north
we go ; but the phenomenon is due to the longer hours of dayhght
throughout the summer, rather than to intensity of illumination or
diJBFerence of temperature. On the other hand, long dayhght, by
prolonging vegetative growth, retards flowering and fruiting; and
late varieties of soya bean may be forced into early ripeness by
artificially shortening their dayhght at midsummer f.

The effect of ultra-violet hght, or any other portion of the
spectrum, is part, and perhaps the chief part, of the same problem.
That ultra-violet hght accelerates growth has been shewn both in
plants and animals f. In tomatoes, growth is favoured by just such
ultra-violet hght as comes very near the end of the solar spectrum §,
and as happens, also, to be especially absorbed by ordinary green-
house glass II . At the other end of the spectrum, in red or orange
light, the leaves become smaller, their petioles longer, the nodes
more numerous, the very cells longer and more attenuated. It is
a physiological problem, and as such it shews how plant-hfe is
adapted, on the whole, to just such rays as the sun sends; but it
also shews the morphologist how the secondary effects of chmate
may so influence growth as to modify both size and form^. An
analogous case is the influence of hght, rather than temperature,
in modifying the coloration of organisms, such as certain butterflies.

* Le chene Zeem d'Ain Draham, Bull, du Directeur General, Tunisie, 1927.

t That the plant grows by turns in darkness and in light, and has its characteristic
growth-phases in each, longer >or shorter according to species and variety and
normal habitat, is a subject now studied under the name of "photoperiodism,"
and become of great practical importance for the northerly extension of cereal
crops in Canada and Russia. Cf. R. G. Whyte and M. A. Oljhovikov, Nature,
Feb. 18, 1939.

I Cf. Kuro Suzuki and T. Hatano, in Proc. Imp. Acad, of Japan, in, pp. 94-96, 1927.
§ Withrow and Benedict, in Bull, of Basic Scient. Research, iii, pp. 161-174, 1931.

II Cf. E. C. Teodoresco, Croissance des plantes aux lumieres de diverses longueurs
d'onde, ^WTi. Sc. Nat, Bat. (8), pp. 141-336, 1929; N. Pfeiffer, Botan. Gaz. lxxxv,
p. 127, 1929; etc.

II See D. T. MacDougal, Influence of light and darkness, etc., Mem. N. Y. Botan.
Garden, 1903, 392 pp.; Growth in trees, Carnegie Inst. 1921, 1924, etc.; J. Wiesner,
Lichtgenuss der Pflunzen, vn, 322 pp., 1907; Earl S. Johnston, Smithson. Misc.
Contrib. 18 pp., 1938; etc. On the curious effect of short spells of light and dark-
ness, see H. Dickson, Proc. E.S. (B), cxv, pp. 115-123, 1938.

242 THE RATE OF GROWTH [ch.

Now if temperature or light affect the rate of growth in strict
uniformity, aHke in all parts and in all directions, it will only lead
to local races or varieties differing in size, as the Siberian goldfinch
or bullfinch differs from our own. But if there be ever so Httle of a
discriminating tendency such as to enhance the growth of one tissue
or one organ more than another*, then it must soon lead to racial,
or even "specific," difference of form.

It is hardly to be doubted that chmate has some such dis-
criminating influence. The large leaves of our northern trees are
an instance of it; and we have a better instance of it still in Alpine
plants, whose general habit is dwarfed though their floral organs
suffer little or no reduction f. Sunhght of itself would seem to be
a hindrance rather than a stimulant to growth; and the familiar
fact of a plant turning towards the sun means increased growth on
the shady side, or partial inhibition on the other.

More curious and still more obscure is the moon's influence on
growth, as on the growth and ripening of the eggs of oysters, sea-
urchins and crabs. Behef in such lunar influence is as old as Egypt;
it is confirmed and justified, in certain cases, nowadays, but the
way in which the influence is exerted is quite unknown J.

Osmotic factors in growth

The curves of growth which we have been studying have a
twofold interest, morphological and physiological. To the morpho-
logist, who has learned to recognise form as a "function of growth,"
the most important facts are these: (1) that rate of growth is an
orderly phenomenon, with general features common to various
organisms, each having its own characteristic rates, or specific
constants; (2) that rate of growth varies with temperature, and so
with season and with climate, and also with various other physical
factors, external and internal to the organism ; (3) that it varies in
different parts of the body, and along various directions or axes:

* Or as we might say nowadays, have a different "threshold value" in one
organ to another.

t Cf. for instance, NageH's classical account of the effect of change of habitat
on alpine and other plants, Sitzungsber. Baier. Akad. Wiss. 1865, pp. 228-284.

X Cf. Munro Fox, Lunar periodicity in reproduction, Proc. R.S. (B), xcv,
pp. 523-550, 1935; also Silvio Ranzi, Pubblic. Slaz. Zool. Napoli, xi, 1931.

Ill] OF OSMOTIC FACTORS IN GROWTH 243

such variations being harmoniously "graded," or related to one
another by a "principle of continuity," so giving rise to the
characteristic form and dimensions of the organism and to the
changes of form which it exhibits in the course of its development.
To the physiologist the phenomenon of growth suggests many other
considerations, and especially the relation of growth itself to chemical
and physical forces and energies.

To be content to shew that a certain rate of growth occurs in
a certain organism under certain conditions, or to speak of the
phenomenon as a "reaction" of the living organism to its environ-
ment or to certain stimuh, would be but an example of that "lack
of particularity" with which we are apt to be all too easily satisfied.
But in the case of growth we pass some little way beyond these
limitations: to this extent, that an affinity with certain types of
chemical and physical reaction has been recognised by a great number
of physiologists*.

A large part of the phenomenon of growth, in animals and still
more conspicuously in plants, is associated with "turgor," that is
to say, is dependent on osmotic conditions. In other words, the
rate of growth depends (as we have already seen) as much or more
on the amount of water taken up into the living cells f, as on the
actual amount of chemical metabohsm performed by them; and
sometimes, as in certain insect-larvae, we can even distinguish
between tissues which grow by increase of cell-size, the result 'of
imbibition, and others which grow by multiplication of their
constituent cells |. Of the chemical phenomena which result in the

* Cf. F. F. Blackman, Presidential Address in Botany, Brit. Assoc, Dublin,
1908. The idea was first enunciated by Baudrimont and St Ange, Recherches sur
le developpement du foetus, Mem. Acad. Sci. xi, p. 469, 1851.

t Cf. J. Loeb, Untersuckungen zur physiologischen Morphologie der Tiere, 1892;
also Experiments on cleavage, Journ. Morphology, vii, p. 253, 1892; Ueber die
Dynamik des tierischen Wachstums, Arch. f. Entw. Mech. xv, p. 669, 1902-3;
Davenport, On the role of water in growth, Boston Soc. N.H. 1897; Ida H. Hyde
in Amer. Journ. Physiology, xii, p. 241, 1905; Bottazzi, Osmotischer Druck und
elektrische Leitungsfahigkeit der Fliissigkeiten der Organismen, in Asher-Spiro's
Ergebnisse der Physiologic, vn, pp. 160-402, 1908; H. A. Murray in Journ. Gener.
Physiology, ix, p. 1, 1925; J. Gray, The role of water in the evolution of the
terrestrial vertebrates, Journ. Exper. Biology, vi, pp. 26-31, 1928; and A. N. J. Heyn,
Physiology of cell-elongation, Botan. Review, vi, pp. 515-574, 1940.

X Cf. C. A. Berger, Carnegie Inst, of Washington, Contributions to Embryology,
xxvu, 1938.

244

THE RATE OF GROWTH

[CH.

actual increase of protoplasm we shall speak presently, but the role
of water in growth deserves a passing word, even in our morpho-
logical enquiry.

The lower plants only Uve and grow in abundant moisture; lew
fungi continue growing when the humidity falls below 85 per cent,
of saturation, and the mould-fungi, such as Penicillium, need more
moisture still (Fig. 75). Their hmit is reached a little below 90%.

Humidity ^

I. 75. Growth of PeniciUium in relation to humidity.

Growth of PeniciUium (at 25° C.) *

Humidity

Growth per hour

(% of saturation)

(mm.)

1000

7-7

97-0

5-0

94-2

1-0

926

0-5

90-8

0-3

Among th^ coelenterate animals growth and ultimate size depend
on Httle more than absorption of water and consequent turgescence,
the process, she wing itself in simple ways. A sea-anemone may live
to ^n immense agef, but its age and size have Httle to do with one

* From R. G. Tomkins, Studies of the growth of moulds, Proc. B.S. (B), cv,
pp. 375-^01, 1929.

t Like Sir John Graham Dalyell's famous "Granny," and Miss Nelson's family
of CereiLS (not Sagartia) of which one still lives at over 80 years old. Cf. J. H.
Ashworth and Nelson Annandale, in Trans. R. Physical Soc. Edin. xxv, pp. 1-14
1904.

Ill] OF TURGESCENCE 245

another. It has an upper limit of size vaguely characteristic of the
species, and if fed well and often it may reach it in a year ; on stinted
diet it grows slowly or may dwindle down; it may be kept at
wellnigh what size one pleases. Certain full-grown anemones were
left untended in war-time, unfed and in water which evaporated
down to half its bulk; they shrank down to little beads, and grew
up again when fed and cared for.

Loeb shewed, in certain zoophytes, that not only must the cells
be turgescent in order to grow, but that this turgescence is possible
only so long as the salt-water in which the cells lie does not overstep
a certain limit of concentration: a limit reached, in the case of
Tubularia, when the salinity amounts to about 3-4 per cent. Sea-
water contains some 3-0 to 3-5 per cent, of salts in the open sea,
but the salinity falls much below this normal, to about 2-2 per cent.,
before Tubularia exhibits its full turgescence and maximal growth;
a further dilution is deleterious to the animal. It is likely enough
that osmotic conditions control, after this fashion, the distribution
and local abundance of many zoophytes. Loeb has also shewn*
that in certain fish-eggs (e.g. of Fundulus) an increasing concentration,
leading to a lessening water-content of the egg, retards the rate of
segmentation and at last arrests it, though nuclear division goes on
for some time longer.

The eggs of many insects absorb water in large quantities, even
doubling their weight thereby, and fail to develop if drought prevents
their doing so ; and sometimes the egg has a thin- walled stalk, or else
a "hydropyle," or other structure by which the water is taken inf.

In the frog, according to Bialaszewicz J, the growth of the embryo
while within the vitelline membrane depends wholly on absorption
of water. The rate varies with the temperature, but the amount
of water absorbed is constant, whether growth be fast or slow.
Moreover, the successive changes of form correspond to definite
quantities of water absorbed, much of which water is intracellular.
The solid residue, as Davenport Has also shewn, may even diminish

* P finger's Archiv, lv, 1893.

t Cf. V. B. Wigglesworth, hised Physiology, 1939, p. 2.

% Beitrage zur Kenntniss d. Wachstumsvorgange bei Amphibienerabryonen, Bull.
Acad. Sci. de Cracovie, 1908, p. 783; also A. Drzwina and C. Bohn, De Taction. . .dea
solutions salines sur les larves des batraciens, ibuL 1906.

246 THE RATE OF GROWTH [ch.

notably, while all the while the embryo continues to grow in bulk
and weight. But later on, and especially in the higher animals, the
water-content diminishes as growth proceeds and age advances;
and loss of water is followed, or accompanied, by retardation and
cessation of growth. A crab loses water as each phase of growth
draws to an end and the corresponding moult approaches; but it
absorbs water in large quantities as soon as the new period of
growth begins*. Moreover, that water is lost as growth goes on has
been shewn by Davenport for the frog, by Potts for the chick, and
particularly by Fehhng in the case of man. Fehhng's results may
be condensed as follows :

Age in weeks (man) 6 17 22 24 26 30 35 39

Percentage of water 97-5 91-8 92-0 89-9 86-4 83-7 82-9 74-2

I

The following illustrate Davenport's results for the frog:

Age in weeks (frog)

1

2

5

7

9

14

41

84

Percentage of water

56-3

58-5

76-7

89-3

931

950

90-2

87-5

The following table epitomises the drying-off of ripening maize f;
it shews how ripening and withering are closely akin, and are but
two phases of senescence (Fig. 76):

Days (from August 6)

22

35

49

56

63

Percentage of water

87

81

77

68

65

58

The bird's egg provides all the food and all the water which the
growing embryo needs, and to carry a provision of water is the
special purpose of the white of the egg ; the water contained in the
albumen at the beginning of incubation is just about what the
chick contains at the end. The yolk is not surrounded by water,
which would diffuse too quickly into it, nor by a crystalloid solution,
whose osmotic value would soon increase; but by a watery albu-
minous colloid, whose osmotic pressure changes slowly as its charge
of water is gradually withdrawn J.

* Cf. A. Krogh, Osmotic regulation in aquatic animals, Cambridge, 1939.
t Henry and Morrison, 1917; quoted by Otto Glaser, on Growth, time and form,
Biolog. Reviews, xiii, pp. 2-58, 1938.

X Cf. James Gray, in Journ. Exper. Biology, iv, pp. 214-225, 1926.

Ill

OF GAIN OR LOSS OF WATER

247

Distribution of water in a hen's egg

Gm.

of water contained in

Day of
incubation

^

Loss by
evaporation

Gain by
combustion

Albumen

Embryo

Yolk

29-9

0-0

8-5

00

00

6

27-2

0-4

8-45

2-4

001

12

20-4

4-6

7-8

5-6

0-27

18

9-2

181

2-3

8-8

1-20

20

2-2

27-4

10

9-8

2-00

The actual amount of water, compared with the dry solids in the
egg, has been determined as follows:

Day of incubation (chick)

5

8

11

14

17

19

Percentage of water

94-7

93-8

92-3

87-7

82-8

82-3

10

20

30 40
Days
Fig. 76. Percentage of water in ripening maize. From Otto Glaser.

We know very httle of the part which all this water plays: how
much is mere "reaction-medium," how much is fixed in hydrated
colloids, how much, in short, is bound or unbound. But we see that
somehow or other water is lost, and lost in considerable amount,
as the embryo draws towards completion and ceases for the time
being to grow.

All vertebrate animals contain much the same amount of water
in their living bodies, say 85 per cent, or thereby, however unequally

248 THE RATE OF GROWTH [ch.

distributed in the tissues that water may be*. Land animals have
evolved from water animals with Httle change in this respect,
though the constant proportion of water is variously achieved.
A. newt loses moisture by evaporation with the utmost freedom, and
regains it by no less rapid absorption through the skin; while a
lizard in his scaly coat is less hable to the one and less capable of
the other, and must drink to replace what water it may lose.

We are on the verge of a difficult subject when we speak of the
role of water in the living tissues, in the growth of the organism,
and in the manifold activities of the cell ; and. we soon learn, among
other more or less unexpected things, that osmotic equihbrium is
neither universal nor yet common in the living organism. The yolk
maintains a higher osmotic pressure than the white of the egg — so
long as the egg is living; and the watery body of a jellyfish, though
aot far off osmotic equilibrium, has a somewhat less salinity than
the sea-water. In other words, its surface acts to some extent as
a semipermeable membrane, and the fluid which causes turgescence
of the tissues is less dense than the sea- water outside!.

In most marine invertebrates, however, the body-fluids con-
stituting the milieu interne are isotonic with the milieu externe, and
vary in these animals pari passu with the large variations to which
sea- water itself is subject. On the other hand, the dwellers in
fresh-water, whether invertebrates or fishes, have, naturally, a more
concentrated medium within than without. As to fishes, diiferent
kinds shew remarkable differences. Sharks and dogfish have an
osmotic pressure in their blood and their body fluids little diiferent

* The vitreous humour is nearly all water, the enamel has next to 'none, the
grey matter has some 86 per cent., the bones, say 22 per cent. ; lung and kidney
take up mor^ than they can hold, and so become excretory or regulatory organs.
Eggs, whether of dogfish, salmon, frogs, ' snakes or birds, are composed, roughly
speaking, of half water and half solid matter.

t Cf. {int. al.) G. Teissier, Sur la teneur en eau. . .de Chrysaora, Bull. Soc. Biol,
de France, 1926, p. 266. And especially A. V. Hill, R. A. Gortner and others. On
the state of water in colloidal and living systems. Trans. Faraday Soc. xxvi,
pp. 678-704, 1930. For recent literature see (e.g.) Homer Smith, in Q. Rev. Biol.
vn, p. 1, 1932; E. K. Marshall, Physiol. Rev. xiv, p. 133, 1934; Lovatt Evans,
Recent Advances in Physiology, 4th ed., 1930; M. Duval, Recherches. . .sur le milieu
interieur des animaux aquatiques, Thhe, Paris, 1925; Paul Portier, Physiologic des
animaux marins. Chap, iii, Paris, 1938; G. P. Wells and I. C. Ledingham, Effects
of a hypotonic environment, Journ. Exp. Biol, xvii, pp. 337-352, 1940.

Ill] OF OSMOTIC REGULATION 249

from that of the sea-water outside: but with certain chemical
diiferences, for instance that the chlorides within are much
diminished, and the molecular concentration is eked out by large
accumulations of urea in the blood. The marine teleosts, on the
other hand, have a much lower osmotic pressure within than that
of the sea-water outside, and only a httle higher than that of their
fresh-water allies. Some, hke the conger-eel, maintain an all but
constant internal concentration, very different from that outside;
and this fish, like others, is constantly absorbing water from the sea ;
it must be exuding or excreting salt continually*. Other teleosts
differ greatly in their powers of regulation and of tolerance, the
common stickleback (which we may come across in a pool or in the
middle of the North Sea) being exceptionally tolerant or "eury-
halinef." Physiology becomes "comparative" when it deals with
differences such as these, and Claude Bernard foresaw the existence
of just such differences: "Chez tous les etres vivants le milieu
interieur, qui est un produit de I'organisme, conserve les rapports
necessaires d'echange avec le milieu exterieur; mais a mesure que
I'organisme devient plus parfait le milieu organique se specifie, et
s'isole en quel que sorte de plus en plus au milieu ambiant J." Claude
Bernard was building, if I mistake not, on Bichat's earUer concept,
famous in its day, of life as "une alternation habituelle d'action de
la part des corps exterieurs,* et de reaction de la part du corps
vivant": out of which grew his still more famous aphorism, "La
vie est I'ensemble des fonctions qui resistent a la mort§."

One crab, like one fish, differs widely from another in its power

* Probably by help of Henle's tubules in the kidney, which structures the dogfish
does not possess. But the gills have their part to play as water-regulators, as
also, for instance, in the crab.

t The grey mullets go down to the sea to spawn, but may live and grow in
brackish or nearly fresh- water. The several species differ much in their adaptability,
and Brunelli sets forth, as follows, the range of salinity which each can tolerate :

M. auratus 24-35 per mille

saliens 16-40

chelo 10-40
capita 5-40

cephalus 4^0

X Introduction d Vetude de la medecine experimentale, 1855, p. 110. For a dis-
cussion of this famous concept see J. Barcroft, "La fixite du milieu interieur est
la condition de la vie libre," Biol. Reviews, viii, pp. 24-87, 1932.

§ Sur la vie et la mart, p. 1.

250 THE RATE OF GROWTH [ch.

of self-regulation; and these physiological differences help to explain,
in both cases, the limitation of this species or that to more or less
brackish, or more or less saline, waters. In deep-sea crabs {Hyas,
for instance) the osnjotic pressure of the blood keeps nearly to that
of the milieu exteme, and falls quickly and dangerously with any
dilution of the latter; but the httle shore-crab (Cardnus moenas)
can hve for many days in sea-water diluted down to one-quarter of
its usual sahnity. Meanwhile its own fluids dilute slowly, but not
near so far; in other words, this crab combines great powers of
osmotic regulation with , a large capacity for tolerating osmotic
gradients which are beyond its power to regulate. How the unequal
balance is maintained is yet but httle understood. But we do know
that certain organs or tissues, especially the gills and the antennary
gland, absorb, retain or ehminate certain elements, or certain ions,
faster than others, and faster than mere diffusion accounts for; in
other words, "ionic*' regulation goes hand in hand with "osmotic"
regulation, as a distinct and even more fundamental phenomenon*.
This at least seems generally true — and only natural — that quickened
respiration and increased oxygen-consumption accompany all such
one-sided conditions: in other words, the "steady state" is only
maintained by the doing of work and the expenditure of energyf.

To the dependence of growth on the uptake of water, and to the
phenomena of osmotic balance and its regulation, HoberJ and also
Loeb were inclined to refer the modifications of form which certain
phyllopod Crustacea undergo when the highly sahne waters which
they inhabit are further concentrated, or are abnormally diluted.
Their growth is retarded by increased concentration, so that
individuals from the more saline waters appear stunted and dwarfish ;
and they become altered or transformed in other ways, suggestive
of "degeneration," or a failure to attain full and perfect develop-

* See especially D. A. Webb, Ionic regulation in Cardnus moenas, Proc. R.S. (B),
cxxix, pp. 107-136, 1940.

t In general the fresh- water Crustacea have a larger oxygen -consumption than
the marine. Stenohaline and euryhaline are terms applied nowadays to species
which are. confined to a narrow range of salinity, or are tolerant of a wide one.
An extreme case of toleration, or adaptability, is that of the Chinese woolly-handed
crab, Eriockeir, which has not only acclimatised itself in the North Sea but has
ascended the Elbe as far as Dresden.

X R. Hober, Bedeutung der Theorie der Losungen fiir Physiologic und Medizin,
Biol. Centralbl. xix, p. 272, 1899.

Ill] OF OSMOTIC REGULATION 251

merit*. Important physiological changes ensue. The consumption
of oxygen increases greatly in the stronger brines, as more and more
active " osmo-regulation " is required. The rate of multiplication is
increased, and parthenogenetic reproduction is encouraged. In the
less sahne waters male individuals, usually rare, become plentiful,
and here the females bring forth their young alive ; males disappear
altogether in the more concentrated brines, and then the females
lay eggs, which, however, only begin to develop when the sahnity
is somewhat reduced.

The best-known case is the little brine-shrimp, Artemia salina,
found in one form or another all the world over, and first discovered
nearly two hundred years ago in the salt-pans at Lymington.
Among many allied forms, one, A. milhausenii, inhabits the natron-
lakes of Egypt and Arabia, where, under the name of "loul," or
"Fezzan-worm," it is eaten by the Arabsf. This fact is interesting,
because it indicates (and investigation has apparently confirmed)
that the tissues of the creature are not impregnated with salt, as
is the medium in which it hves. In short Artemia, hke teleostean
fishes in the sea, hves constantly in a "hypertonic medium"; the
fluids of the body, the milieu interne, are no more salt than are those
of any ordinary crustacean or other animal, but contain only some
0-8 per cent, of NaCl J , while the milieu externe may contain from
3 to 30 per cent, of this and other salts; the skin, or body- wall, of.
the creature acts as a "semi-permeable membrane," through which
the dissolved salts are not permitted to diffuse, though water passes
freely. When brought into a lower concentration the animal may
grow large and turgescent, until a statical equilibrium, or steady
state, is at length attained.

Among the structural changes which result from increased con-

* Schmankewitsch, Zeitschr. f. wiss. Zool. xxix, p. 429, 1877. Schraankewitsch
has made equally interesting observations on change of size and form in other
organisms, after some generations in a milieu of altered density ; e.g. in the flagellate
infusorian Ascinonema acinus Biitschli.

t These "Fezzan- worms," when first described, were supposed to be "insects'
eggs"; cf. Humboldt^ Personal Narrative, vi, i, 8, note; Kirby and Spence, Letter x.

X See D. J. Kuenen, Notes, systematic and physiological, on Artemia, Arch.
N4erland. Zool. iii, pp. 365-449, 1939; cf. also Abonyi, Z.f. w. Z. cxiv, p. 134, 1915.
Cf. Mme. Medwedewa, Ueber den osmotischen Druck der Haemolymph v. Artemia;
in Ztsch. f. vergl. Physiolog. v, pp. 547-554, 1922.

252

THE RATE OF GROWTH

[CH.

centration of the brine (partly during the hfe-time of the individual,
but more markedly during the short season which suffices for the
development of three or four, or perhaps more, successive genera-
tions), it is found that the tail comes to bear fewer and fewer
bristles, and the tail-fins themselves tend at last to disappear:
these changes corresponding to what have been described as the
specific characters of A. milhausenii, and of a still more extreme
form, A. koppeniana] while on the other hand, progressive dilution
of the water tends ' to precisely opposite conditions, resulting in
forms which have also been described as separate species, and even

u

^wwwWWWH
I

I

Artewia s.str.

Callaonella

Fig. 77. Brine-shrimps (Artemia), from more or less saline water. Upper figures
shew tail-segment and tail-fins; lower figures, relative length of cephalothorax
and abdomen. After Abonyi.

referred to a separate genus, Callaonella, closely akin to Branchipus
(Fig. 77). Pari passu with these changes, there is a marked change
in the relative lengths of the fore and hind portions of the body,
that is to say, of the cephalothorax and abdomen: the latter
growing relatively longer, the Salter the water. In other words,
not only is the rate of growth of the whole animal lessened by the
sahne concentration, but the specific rates of growth in the parts
of its body are relatively changed. This latter phenomenon lends
itself to numerical statement, and Abonyi has shewn that we may
construct a very regular curve, by plotting the proportionate length
of the creature's abdomen against the salinity, or density, of the
water; and the several species of Artemia, with all their other
correlated specific characters, are then found to occupy successive,
more or less well-defined, and more or less extended, regions of the

Ill]

OF THE BRINE-SHRIMPS

253

curve (Fig. 78). In short, the density of the water is so clearly
"specific," that we might briefly define Artemia jelskii, for instance,
as the Artemia of density 1000-1010 (NaCl), or all but fresh water,
and the typical A. salina (or principalis) as the Artemia of density
1018-1025, and so on*.

Koppeniana
160 """^

140

120

100

1000

1020

1080

1100

1040 1060

Density of water
Fig. 78, Percentage ratio of length of abdomen to cephalothorax
in brine -shrimps, at various salinities. After Abonyi.

These Artemiae are capable of living in waters not only of great
density, but of very varied chemical composition, and it is hard to
say how far they are safeguarded by semi-permeabihty or by specific
properties and reactions of the living colloids "j". The natron-lakes,

* Different authorities have recognised from one to twenty species of Artemia.
Daday de Dees {Ann. sci. nat. 1910) reduces the salt-water forms to one species
with four varieties, but keeps A. jelskii in a separate sub-genus. Kuenen suggests
two species, A. salina and gracilis, one for the European and one for the American
forms. According to Schmankewitsch every systematic character can be shewn
to vary with the external medium. Cf. Professor Labbe on change of characters,
specific and even generic, of Copepods according to the ^H of saline waters at
Le Croisic, Nature, March 10, 1928.

t We may compare Wo. Ostwald's old experiments on Daphnia, which died in
a pure solution of NaCl isotonic with normal sea-water. Their death was not to
be explained on osmotic grounds; but was seemingly due to the fact that the
organic gels do not retain their normal water- content save in the presence of such
concentrations of MgClj (and other salts) as are present in sea-water.

254 THE RATE OF GROWTH [ch.

for instance, contain large quantities of magnesium sulphate; and
the Artemiae continue to live equally well in artificial solutions
where this salt, or where calcium chloride, has largely replaced the
common salt of the more usual habitat. Moreover, such waters as
those of the natron-lakes are subject to great changes of chemical
composition as evaporation and concentration proceed, owing to the
different solubilities of the constituent salts; but it appears that
the forms which the Artemiae assume, and the changes which they
undergo, are identical, or indistinguishable, whichever of the above
salts happen to exist or to predominate in their saline habitat. At
the same time we still lack, so far as I know, the simple but crucial
experiments which shall tell us whether, in solutions of different
chemical composition, it is at equal densities, or at isotonic concen-
trations (that is to say, under conditions where the osmotic pressure,
and consequently the rate of diffusion, is identical), that the same
changes of form and structure are produced and corresponding
phases of equihbrium attained.

Sea-water has been described as an instance of the "fitness of the
environment*" for the maintenance of protoplasm in an appropriate
milieu; but our Artemias suffice to shew how nature, when hard
put to it, makes shift with an environment which is wholly abnormal
and anything but "fit."

While Hober and others f have referred all these phenomena to
osmosis, Abonyi is inclined to believe that the viscosity, or
mechanical resistance, of the fluid also reacts upon the organism;
and other possible modes of operation have been suggested. But
we may take it for certain that the phenomenon as a whole is not
a simple one. We should have to look far in organic nature for
what the physicist would call simple osmosis % ; and assuredly there
is always at work, besides the passive phenomena of intermolecular

* L. H. Henderson, The Fitness of the Environment, 1913.

t Cf. Schmankewitsch, Z. f. w. Zool. xxv, ISTi); xxix, 1877,. etc.; transl. in
appendix to Packard's Monogr. of N. American Phyllopoda, 1^83, pp. 466-514;
Daday de Dees, Ann. Sci. Nat. (Zool), (9), xi, 1910; Samter und Heymons, Abh.
d. K. pr. Akad. Wiss. 1902; Bateson, Mat. for the Study of Variation, 1894, pp.
96-101; Anikin, Mitlh. Kais. Univ. Tomsk, xiv: Zool. Centralhl. vi, pp. 756-760,
1908; Abonyi, Z.f. w. Zool. cxiv, pp. 96-168, 1915 (with copious bibliography), etc.

% Cf. C. F. A. Pantin, Body fluids in animals, Biol. Reviews, \i, p. 4, 1931;
J. Duclaux, Chimie apjMquee a la biologic, 1937, ii, chap. 4.

Ill] OF CATALYTIC ACTION 255

diffusion, some other activity to play the part of a regulatory
mechanism*.

On growth and catalytic action

In ordinary chemical reactions we have to deal (1) with a specific
velocity proper to the particular reaction, (2) with variations due
to temperature and other physical conditions, (3) with variations
due to the quantities present of the reacting substances, according
to Van't Hoff's "Law of Mass Action," and (4) in certain cases with
variations due to the presence of "catalysing agents," as BerzeHus
called them a hundred years agof. In the simpler reactions, the
law of mass involves a steady slo wing-down of the process as the
reaction proceeds and as the initial amount of substance diminishes:
a phenomenon, however, which is more or less evaded in the organism,
part of whose energies are devoted to the continual bringing-up of
supphes.

Catalytic action occurs when some substance, often in very
minute quantity, is present, and by its presence produces or
accelerates a reaction by opening "a way round," without the
catalysing agent itself being diminished or used up J. It diminishes
the resistance somehow — little as we know what resistance means

* According to the empirical canon of physiology, that, as Leon Fredericq
expresses it (Arch, rfc Zool. 1885), '*L'etre vivant est agence de telle maniere que
chaque influence pertyrbatrice provoque d'elle-meme la mise en activite de Tappareil
compensateur qui doit neutraliser et reparer le dommage." Herbert Spencer had
conceived a similar principle, and thought he recognised in it the vis medicutrix.
Nahirae. It is the physiological analogue of the "principle of Le Chatelier " (1888),
with this important difference that the latter is a rigorous and quantitative law,
ba8e<i on a definite and stable equilibrium. The close relation between the two is
maintained by Le Dantec {La titabilite de la Vie, 1910, p, 24), and criticised by
Lotka {Physical Biology, p, 283 seq.).

t In a paper in the Berliner Jahrbuch for 1836, This paper was translated in
the Edinburgh New Philosophical Journal in the following year; and a curioas
little paper On the coagulation of albumen, and catalysis, by Dr Samuel Brown,
followed in the Edinburgh Academic Annual for 1840,

X Such phenomena come precisely under the head of what Bacon called
Instances of Magic: "By which I mean those wherein the material or efficient
cause is scanty and small as compared with the work or effect produced; so that
even when they are common, they seem like miracles, some at first sight, others
even after attentive consideration. These magical effects are brought about in
three ways. . .[of which one is] by excitation or invitaticm in another body, as in
the magnet which excites numberless needles without losing any of its virtue, or
in yeast and such-like." Nov. Org,, cap. li.

256 THE RATE OF GROWTH [ch.

in a chemical reaction. But the velocity-curve is not altered in
form ; for the amount of energy in the system is not affected by the
presence of the catalyst, the law of mass exerts its eifect, and the
rate of action gradually slows down. In certain cases we have
the remarkable phenomenon that a body capable of acting as a
catalyser is necessarily formed as a product, or by-product, of the
main reaction, and in such a case as this the reaction- velocity will
tend to be steadily accelerated. Instead of dwindhng away, such
a reaction continues with an ever-increasing velocity: always
subject to the reservation that limiting conditions will in time make
themselves felt, such as a failure of some necessary ingredient (the
"law of the minimum"), or the production of some substance which
shall antagonise and finally destroy the original reaction. Such an
action as this we have learned, from Ostwald, to describe as "auto-
catalysis." Now we know that certain products of protoplasmic
metabohsm — we call them enzymes — are very powerful catalysers,
a fact clearly understood by Claude Bernard long ago*; and we
are therefore entitled, to that extent, to speak of an autocatalytic
action on the part of protoplasm itself.

Going a httle farther in the footsteps of Claude Bernard, Chodat
of Geneva suggested (as we are told by his pupil Monnier) that
growth itself might be looked on as a catalytic, or autocatalytic
reaction: "On peut bien, ainsi que M. Chodat I'a propose, considerer
I'accroissement comme une reaction chimique complexe, dans
laquelle le catalysateur est la cellule vivante, et les corps en presence
sont I'eau, les sels et Facide carboniquet-"

A similar suggestion was made by Loeb, in connection with the

* "Les diastases contiennent, en definitive, le secret de la vie. Or, les actions
diastatiques nous apparaissent comme des phenomenes catalytiques, en d'autres
termes, des accelerations de vitesse de reaction." Cf. M. F. Porchet, Rewie
Scientifique, 18th Feb. 1911. For a last word on this subject, see W. Frankenberger,
Katalytische Umsetzungen in homogenen u. enzymatischen Systemen, Leipzig, 1937.

t Cf. R. Chodat, Principes de Botanique (2nd ed.), 1907, p. 133; A. Monnier, La
loi d'accroissement des vegetaux, Publ. de VInst. de Bot. de VUniv. de Geneve (7),
m, 1905. Cf. W. Ostwald, Vorlesungen iiber Naturphilosophie, 1902, p. 342;
Wo. Ostwald, Zeitliche Eigenschaften der Entwicklungsvorgange, in Roux's
Vortrdge, Heft 5, 1908; Robertson, Normal growth of an individual, and its
biochemical significance, Arch.f. Entw. Mech. xxv, pp. 581-614; xxvi, pp. 108-118,
1908; S. Hatai, Growth-curves from a dynamical standpoint, Anat. Record, v,
p. 373, 1911; A. J. Lotka, Ztschr. f. physikal. Chemie, Lxxn, p. 511, 1910; lxxx,
p. 159, 1912; etc.

Ill] OF AUTOCATALYSIS 257

synthesis of nuclear protoplasm, or nuclein; for he remarked that,
as in an autocatalysed chemical reaction the rate of synthesis
increases during the initial stage of cell-division in proportion to the
amount of nuclear matter already there. In other words, one of
the products of the reaction, i.e. one of the constituents of the
nucleus, accelerates the production of nuclear from cytoplasmic
material. To take one more instance, Blackman said, in the address
already quoted, that "the botanists (or the zoologists) speak of
growth, attribute it to a specific power of protoplasm for assimila-
tion, and leave it alone as a fundamental phenomenon; but they
are much concerned as to the distribution of new growth in innu-
merable specifically distinct forms. While the chemist, on the
other hand, recognises it as a famihar phenomenon, and refers it to
the same category as his other known examples of autocatalysis."

Later on, Brailsford Robertson upheld the autocatalytic theory
with skill and learning*; and knowing well that growth was no
simple solitary chemical reaction, he thought that behind it lay some
one master-reaction, essentially autocatalytic, by which protoplasmic
synthesis was effected or controlled. He adduced at least one
curious case, in the growth and multiphcation of the Infusoria,
which can hardly be described otherwise than as catalytic. Two
minute individuals (of Enchelys or Colpodiuyn) kept in the same drop
of water, so enhance each other's rate of asexual reproduction that
it may be many times as great when two are together as when one
is alone; the phenomenon has been called allelocatalysis. When a
single infusorian is isolated, it multiplies the quicker the smaller the
drop it is in — a further proof or indication that something is being
given oif, in this instance by the living cells, which hastens growth
and reproduction. But even the ordinary multiplication of a
bacterium, which doubles its numbers every few minutes till (were
it not for hmiting factors) those numbers would be all but incal-
culable in a day, looks like and has been cited as a simple but most
striking instance of the potentiahties of protoplasmic catalysis.

It is not necessary for us to pursue this subject much further.

* T. B. Robertson, The Chemical Basis of Growth and Senescence, 1923; and
earlier papers. Cf. his Multiplication of isolated infusoria, Biochem. Journ. xv,
pp. 598-611, 1921; cf. Journ. Physiol, lvi, pp. 404-412, 1921; R. A. Peters,
Substances needed for the growth of. . .Colpodiy,m, Journ. Physiol, lv, p. 1, 1921.

258 THE RATE OE GROWTH [ch.

It is sufficiently obvious that the normal S-shaped curve of growth
of an organism resembles in its general features the velocity-curve
of chemical autocatalysis, and many writers have enlarged on the
resemblance; but the S-shaped curve of growth of a population
resembles it just as well. When the same curve depicts the growth
of an individual, and of a population, and the velocity of a chemical
reaction, it is enough to shew that the analogy between these is a
mathematical and not a physico-chemical one. The sigmoid curve
of growth, common to them all, is sufficiently explained as an
interference effect, due to opposing factors such as we may use a
differential equation to express : a phase of acceleration is followed
by a phase of retardation, and the causes of both are in each case
complex, uncertain or unknown. . Nor are points of difference lacking
between the chemical and the biological phenomena. As the
chemical reaction draws to a close, it is by the gradual attainment
of chemical equihbrium; but when organic growth comes to an end,
it is (in all but the lowest organisms) by reason of a very different
kind of equilibrium, due in the main to the gradual differentiation
of the organism into parts, among whose pecuHar properties or
functions that of growth or multiphcation falls into abeyance.

The analogy between organic growth and chemical autocatalysis
is close enough to let us use, or try to use, just such mathematics as
the chemist applies to his reactions, and so to reduce certain curves
of growth to logarithmic formulae. This has been done by many, and
with no httle success in simple cases. So have we done, partially,
in the case of yeast ; so the statisticians and actuaries do with human
populations; so we may do again, borrowing (for illustration) a
certain well-known study of the growing sunflower (Figs. 79, 80).
Taking our mathematics from elementary physical chemistry, we
learn that :

The velocity of a reaction depends on the concentration a of
the substance acted on: V varies as a,

V = Ka.

The concentration continually decreases, so that at time t (in a
monomolecular reaction),

in

OF AUTOCATALYSIS

259

4 5 6 7
Time, in weeks

10 II

Fig. 79. Growth of sunflower-stem : observed and calculated curves.
From Reed and Holland.

cms.

250

200

150

100

50

■ -■ ■ . *

• ^ —

• >^

/

/Calculauted (autocaialytic)

/ curve

• /

•/

y^\ 1 1 1

1 1 1 1 1 1 t 1

Weeks 1

10 li 12

Fig. 80. Growth of sunflower-stem : calculated (autocatalytic) curve.
After Reed and Holland.

260 THE RATE OF GROWTH [ch.

But if the substance produced exercise a catalytic effect, then the
velocity will vary not only as above but will also increase as x
increases: the equation becomes

V = -T- ^k'x(a — x),

Cut

which is the elementary equation of autoca.talysis. Integrating,

at a — X

In our growth-problem it is sometimes found convenient to choose
for our epoch, t', the time when growth is half-completed, as the
chemist takes the time at which his reaction is half-way through;
and we may then write (with a changed constant)

This is the physico-chemical formula which Reed and Holland
apply to the growing sunflower-stem — a simple case*. For a we
take the maximum height attained, viz. 254-5 cm. ; for t\ the epoch
when one-half of that height was reached, viz. (by interpolation)
about 34-2 days. Taking an observation at random, say that for
the 56th day, when the stem was 228-3 cm. high, we have

K in this case is found to be 0-043, and the mean of all such
determinations t is not far difierent.

Applying this formula to successive epochs, we get a calculated
curve in close agreement with the observed one; and by well-
known statistical methods we confirm, and measure, its "closeness
of fit." But jtist as the chemist must vary and develop his funda-
mental formula to suit the course of more and more comphcated
reactions, so the biologist finds that only the simplest of his curves

* H. S. Reed and R. H. Holland, The growth-rate of an annual plant, Helianthus,
Proc. Nat. Acad, of Sci. (Washington), v, p. 135, 1919; cf. Lotka, op. cit., p. 74,
A sifnilar case is that of a gourd, recorded by A. P. Anderson, Bull. Survey,
Minnesota, 1895, and analysed by T. B. Robertson, ibid. pp. 72-75.

t Better determined, especially in more complex cases, by the method of least
squares.

Ill] ITS CHEMICAL ASPECT 261

of growth, or only portions of the rest, can be fitted to this simplest
of formulae. In a Hfe-time are many ages; and no all-embracing
formula covers the infant in the womb, the suckling child, the
growing schoolboy, the old man when his work is done. Besides,
we need such a formula as a biologist can understand ! One which
gives a mere coincidence of numbers may be of little use or none,
unless it go some way to depict and explain the modus operandi of
growth. As d'Ancona puts it : "II importe d'apphquer des formules
qui correspondent non seulement au point de vue geometrique, mais
soient representees par des valeurs de signification biologique."
A mere curve-diagram is better than an empirical formula; for it
gives us at least a picture of the phenomenon, and a qualitative
answer to the problem.

Growth of sunflower-stem. (After Reed and Holland)

1st diff.

15-8

24-4

33-3

39-2

38-4

31-6

22-6

14-4

8-5

4-9

2-8

The chemical aspect of growth

As soon as we touch on such matters as the chemical phenomenon
of catalysis we are on the threshold of a subject which, if we were
able to pursue it, would lead us far into the special domain of
physiology ; and there it would be necessary to follow it if we were
dealing with growth as a phenomenon in itself, instead of mainly
as a help to our study and comprehension of form. The whole
question of diet, of overfeeding and underfeeding*, would present

* For example, A. S. Parker has shewn that mice suckled by rats, and conse-
quently much overfed, grow so quickly that in three weeks they reach double their
normal weight; but their development is not accelerated; Ann. Appl. Biol, xvi,
1929.

Height ((

cm.)

j^

> (days)

Observed

Calculated

7

17-9

21-9

14

34-4

37-7

21

67-8

62-1

28

981

95-4

35

1310

134-6

42

1690

173-0

49

205-5

204-6

56

228-3

227-2

63

247-1

241-6

70

250-5

250-1

77

253-8

255-0

84

254-5

257-8

262 THE RATE OF GROWTH [ch.

itself for discussion*. But without opening up this large subject,
we may say one more passing word on the remarkable fact that
certain chemical substances, or certain physiological secretions,
have the power of accelerating or of retarding or in some way
regulating growth, and of so influencing the morphological features
of the organism.

To begin with there are numerous elements, such as boron,
manganese, cobalt, arsenic, which serve to stimulate growth, or
whose complete absence impairs or hampers it; just as there are
a few others, such as selenium, whose presence in the minutest
quantity is injurious or pernicious. The chemistry of the hving
body is more complex than we were wont to suppose.

Lecithin was shewn long ago to have a remarkable power of
stimulating growth in animals t, and accelerators of plant-growth,
foretold by Sachs, were demonstrated by Bottomley and others J;
the several vitamins are either accelerators of growth, or are indis-
pensable in order that it may proceed.

In the little duckweed of our ponds and ditches [Lemna minor) the botanists
have found a plant in which growth and multiplication are reduced to very
simple terms. For it multiplies by budding, grows a rootlet and two or three
leaves, and buds again; it is all young tissue, it carries no dead load; while
the sun shines it has no lack of nourishment, and may spread to the limits of
the pond. In one of Bottomley's early experiments, duckweed was grown
(1) in a "culture solution" without stint of space or food, and (2) in the same,
with the addition of a little bacterised peat or "auximone." In both cases the
little plant spread freely, as in the first, or Malthusian, phase of a population
curve; but the peat greatly accelerated the rate, which was not slow before.
Without the auximone the population doubled in nine or ten days, and with
it in five or six; but in two months the one was seventy-fold the other !

The subject has grown big from small beginnings. We know
certain substances, haematin being one, which stimulate the growth
of bacteria, and seem to act on them as true catalysts. An obscure
but complex body known as "bios" powerfully stimulates the
growth of yeast; and the so-called auxins, a name which covers
numerous bodies both nitrogenous and non-nitrogenous, serve in

* For a brief resume of this subject see Morgan's Experimental Zoology, chap. xvi.

t Hatai, Amer. Joum. Physiology, x, p. 57, 1904; Danilewsky, C.R. cxxi, cxxn,
1895-96.

X W. B. Bottomley, Proc. R.S. (B), lxxxviii, pp. 237-247, 1914, and other
papers. O. Haberlandt, Beitr. z. allgem. Botanik, 1921.

Ill] OF HORMONES 263

minute doses to accelerate the growth of the higher plants*. Some
of these "growth-substances" have been extracted from moulds or
from bacteria, and one remarkable one, to which the name auxin
is especially applied, from seedhng oats. This last is no enzyme
but a stable non-nitrogfenous substance, which seems to act by
softening the cell- wall and so facihtating the expansion of the cell.
Lastly the remarkable discovery has been made that certain indol-
compounds, comparatively simple bodies, act to all intents and
purposes in the same way as the growth-hormones or natural
auxins, and one of these "hetero-auxins." an indol-acetic acidf,
is already in common and successful use to promote the growth and
rooting of cuttings.

Growth of duckweed, with and without peat-auodmone

Without With

^

^

eeks Obs.

Calc.

Obs.

Calc.

20

20

20

20

1 30

33

38

55

2 52

54

102

153

.1 77

88

326

424

4 135

155

1,100

1,173

5 211

237

3,064

3,250

6 326

390

6,723

8,980

7 550

640

19,763

2,490

8 1052

1048

69,350

68,800

Percentage increase,

164 o/o

2770/0

per week

There are kindred matters not less interesting to the morphologist.
It has long been known that the pituitary body produces, in its
anterior lobe, a substance by which growth is increased and regulated.
This is what we now call a "hormone" — a substance produced in
one organ or tissue and regulating the functions of another. In this
case atrophy of the gland leaves the subject a dwarf, and its hyper-

* The older literature is summarised by Stark, Ergebn. d. Biologies n, 1906;
the later by N. Nielsen, Jh. wiss. Botan. Lxxm, 1930; by Boyson Jensen, Die
Wuchsstojftheorie, 1935; by F. W. Went and K. V. Thimann, Phytohormones, New
York, 1937, and by H. L. Pearse, Plant hormones and their practical importance.
Imp. Bureau of Horticulture, 1939. Cf. Went, Bee. d. Trav. Botan. Neerl. xxv, p. 1,
1928; A. N. J. Heyn, ibid, xxviii, p. 113, 1931.

t Discovered by Kogl and Kostermans, Ztschr. f. physiol. Chem. ccxxxv, p. 201,
1934. Cf. {int. al.) P. W. Zimmermann and F. W. Wilcox in Contrib. Boyce-
Thompson Instil. 1935.

264 THE RATE OF GROWTH 264

trophy or over-activity goes to the making of a giant; the Umb-
bones of the giant grow longer, their epiphyses get thick and clumsy,
and the deformity known as "acromegaly" ensues*. This has
become a famihar illustration of functional regulation, by some
glandular or "endocrinal" secretion, some enzyme or harmozone
as Gley called it, or hormone'f as Bayliss and Starhng called it — in
the particular case where the function to be regulated is growth,
with its consequent influence on form. But we may be sure that
this so-called regulation of growth is no simple and no specific thing,
but imphes a far-reaching and complicated influence on the bodily
metabohsmj.

Some say that in large animals the pituitary is. apt to be dispro-
portionately large §; and the giant dinosaur Branchiosaurus, hugest
of land animals, is reputed to have the largest hypophyseal recess
(or cavity for the pituitary body) ever observed.

The thyroid also has its part to play in growth, as Gudernatsch
was the first to shew||; perhaps it acts, as Uhlenhorth suggests,
by releasing the pituitary hormone. In a curious race of dwarf
frogs both thyroid and pituitary were found to be atrophied ^. When
tadpoles are fed on thyroid their legs grow out long before the usual
time; on the other hand removal of the thyroid delays metamor-
phosis, and the tadpoles remain tadpoles to an unusual size**.

The great American bull-frog (R. Catesheiana) fives for two or
three years in tadpole form; but a diet of thyroid turns the little
tadpoles into bull-frogs before they are a month old If- The converse

* Cf. E. A. Schafer, The function ofjthe pituitary body, Proc. R.S. (B), lxxxi,
p. 442, 1904.

t It is not easy to dtaw a line between enzyme and vitamin, or between hormone
and enzyme.

X The -physiological relations between insulin and the pituitary body might
seem to indicate that it is the carbohydrate metabolism which is more especially
concerned. Cf. (e.g.) Eric Holmes, Metabolism of the Living Tissue, 1937.

§ Van der Horst finds this to be the case in Zalophus and in the ostrich, compared
with smaller seals or birds; cf. Ariens Kappers, Journ. Anat. lxiv, p. 256, 1930.

II Gudernatsch, in Arch. f. Entw. Mech. xxxv, 1912.

<;; Eidmann, ibid, xlix, pp. 510-537, 1921.

** Allen, Journ. Exp. Zod. xxiv, p. 499, 1918. Cf. [int. al.) E. Uhlenhuth,
Experimental production of gigantism, Journ. Gen. Physiol, ill, p. 347; iv, p. 321,
1921-22.

tt W. W. Swingle, Journ. Exp. Zool. xxiv, 1918; xxxvn, 1923; Journ. Gen.
Physiol. I, II, 1918-19; etc!

Ill] THE THYROID GLAND . 265

experiment has been performed on ordinary tadpoles*; with their
thyroids removed they remain normal to all appearance, but the
weeks go by and metamorphosis does not take place. Gill-clefts
and tail persist, no limbs appear, brain and gut retain their larval
features; but months after, or apparently at any time, the belated
tadpoles respond to a diet of thyroid, and may be turned into frogs
by means of it. The Mexican axolotl is a grown-up tadpole which,
when the ponds dry up (as they seldom do), completes its growth
and turns into a gill-less, lung-breathing newt or salamander!; but
feed it on thyroid, even for a single meal, and its metamorphosis is
hastened and ensured {.

Much has been done since these pioneering experiments, all going
to shew that the thyroid plays its active part in the tissue-changes
which accompany and constitute metamorphosis. It looks as though
more thyroid meant more respiratory activity, more oxygen-
consumption, more oxidative metabohsm, more tissue-change, hence/
earlier bodily development §. Pituitary and thyroid are very different
things; the one enhances growth, the other retards it. Thyroid
stimulates metabolism and hastens development, but the tissues
waste.

It is a curious fact, but it has often been observed, that starvation
or inanition has, in the long run, a similar effect of hastening
metamorphosis II . The meaning of this phenomenon is unknown.

An extremely remarkable case is that of the "galls", brought
into existence on various plants in response to the prick of a small
insect's ovipositor. One tree, an oak for instance, may bear galls

* Bennett Allen, Biol. Bull, xxxii, 1917; Journ. Exp. Zool. xxiv, 1918; xxx,
1920; etc.

t Colorado axolotls are much more apt to metamorphose than the Mexican
variety.

J Babak, Ueber die Beziehung der Metamorphose . . . zur inneren Secretion,
Centralbl f. Physiol, x, 1913. Cf. Abderhalden, Studien iiber die von einzelrien
Organen hervorgebrachten Substanzen mit spezifischer Wirkung, Pfliiger's Archiv,
CLXii, 1915.

§ Certain experiments by M. Morse {Journ. Biol. Chem. xix, 1915) seeihed to
shew that the effect of thyroid on metamorphosis depended on iodine; l;^t the
case is by no means clear (cf. 0. Shinryo, Sci. Rep. Tohoku Univ. iii, 1928, and
others). The axolotl is said to shew little response to experimental iodine, and
its ally Necturus none at all (cf. B. M. Allen, in Biol. Reviews, xiii, 1939).

!| Cf. Krizensky, Die beschleunigende Ein wirkung des Hungerns auf die Meta-
morphose, Biol. Centralbl. xi.iv, 1914. Cf. antea, p. 170.

266 THE RATE OF GROWTH [ch.

of many kinds, well-defined and widely different, each caused to
grow out of the tissues of the plant by a chemical stimulus contri-
buted by the insect, in very minute amount; and the insects are
so much alike that the galls are easier to distinguish than the flies.
The same insect may produce the same gall on different plants,
for instance on several species of willow ; or sometimes on different
parts, or tissues, of the same plant. Small pieces of a dead larva
have been used to infect a plant, and a gall of the usual kind has
resulted. Beyerinck killed the eggs with a hot wire as soon as
they were deposited in the tree, yet the galls grew as usual. Here,
as Needham has lately pointed out, is a great field for reflection
and future experiment. The minute drop of fluid exuded by the
insect has marvellous properties. It is not only a stimulant of
growth, like any ordinary auxin or hormone; it causes the growth
of a peculiar tissue, and shapes it into a new and specific form*.

Among other illustrations (which are plentiful) of the subtle
influence of some substance upon growth, we have, for instance,
the growth of the placental decidua, which Loeb shewed to be due
to a substance given off by the corpus luteum, lending to the uterine
tissues an enhanced capacity for growth, to be called into action by
contact with the ovum or even of a foreign body. Various sexual
characters, such as the plumage, comb and spurs of the cock, arise
in hke manner in response to an internal secretion or "male
hormone " ; and when castration removes the source of the secretion,
well-known morphological changes take place. When a converse
change takes place the female acquires, in greater or less degree,
characters which are proper to the male: as in those extreme cases,
known from time immemorial, when an old and barren hen assumes
the plumage of the cockf.

The mane of the lion, the antlers of the stag, the tail of the peacock,
are all examples of intensifled differential growth, or localised and

* Joseph Needham, Aspects nouveaux de la chimie et de la biologic de la croia-
sance organisee. Folia Morphologica, Warszawa, viii, p. 32, 1938. On galls, see
{int. al.) Cobbold, Ross und Hedicke, Die Pflanzengallen, Jena, 1927; etc. And
on their "raorphogenic stimulus", cf. Herbst, Biolog. Cblt., 1894-5, passim.

t The hen which assumed the voice and plumage of the male was a portent or
omen — gallina cecinit. The first scientific account was John Hunter's celebrated
Account of an extraordinary pheasant, and Of the appearance of the change
of sex in Lady Tynte's peahen, Phil. Trans, lxx, pp. 527, 534, 1780.

Ill]

THE MALE HORMONES

267

sex-linked hypertrophy; and in the singular and striking plumage
of innumerable birds we may easily see how enhanced growth of a
tuft of feathers, perhaps exaggeration of a single plume, is at
the root of the whole matter. Among extreme instances we may
think of the immensely long first primary of the pennant-winged
nightjar; of the long feather over the eye in Pteridophora alberti,

Fig. 81. A single pair of hypertrophied feathers in a bird-of-paradise,
Pteridophora alberti.

.-I

Fig. 82. Unequal growth in the three pairs of tail-feathers of a humming-bird
{Loddigesia). 1, rudimentafy: 2, short and stiff; 3, long and spathulate.

or the Fix long plumes over or behind the eye in the six-shafted
bird-of-paradise; or among the humming-birds, of the long outer
rectrix in Lesbia, the second outer one in Aethusa, or of the extra-
ordinary inequalities of the tail-feathers of Loddigesia mirabilis,
some rudimentary, some short and straight and stiff, and other two
immensely elongated, curved and spathulate. The sexual, hormones
have a potent influence on the plumage of a>bird ; they serve, somehow,
to orientate and regulate the rate of growth from one feather-tract
to another, and from one end to another, even from one side to the
other, of a single feather. An extreme case is the occasional pheno-

268 THE RATE OF GROWTH [ch.

menon of a " gynandrous " feather, male and female on two sides
of the same vane*.

While unequal or differential growth is of pecuhar interest to
the morphologist, rate of growth pure and simple, with all the
agencies which control or accelerate it, remains of deeper importance
to the practical man. The live-stock breeder keeps many desirable
quahties in view: constitution, fertility, yield and quahty of milk
or wool are some of these; but rate of growth, with its corollaries
of early maturity and large ultimate size, is generally more important
than them all. The inheritance of size is somewhat complicated,
and limited from the breeder's point of view by the mother's
inability to nourish and bring forth a crossbred offspring of a breed
larger than her own. A cart mare, covered by a Shetland sire,
produces a good-sized foal; but the Shetland mare, crossed with
a carthorse, has a foal a little bigger, but not much bigger, than
herself (Fig. 83). In size and rate of growth, as in other qualities,
our farm animals differ vastly from their wild progenitors, or from
the " un-improved " stock in days before Bake well and the other
great breeders began. The improvement has been brought about
by "selection"; but what lies behind? Endocrine secretions,
especially pituitary, are doubtless at work; and already the stock-
raiser and the biochemist may be found hand in hand.

If we once admit, as we are now bound to do, the existence of
factors which by their physiological activity, and apart from any
direct action of the nervous system, tend towards the acceleration
of growth and consequent modification of form, we are led into wide
fields of speculation by an easy and a legitimate pathway. Professor
Gley carries such speculations a long, long way: for He saysf that
by these chemical influences "Toute une partie de la construction
des etres parait s'expliquer d'une fayon toute mecanique. La forteresse,
si longtemps inaccessible, du vitahsme est entamee. Car la notion
morphogenique etait, suivant le mot de Dastre J , comme ' le dernier
reduit de la force vitale'."

* See an interesting paper by Frank R, Lillie and Mary Juhn, on The physiology
of development of feathers: I, Growth-rate and pattern in the individual feather.
Physiological Zoology, v, pp. 124-184, 1932, and many papers quoted therein.

f Le Neo-vitalisme, Revue Scientifique, March 1911.

X La Vie et la Mart, 1902, p. 43.

Ill]

OF INHERITANCE OF SIZE

269

The physiological speculations we need not discuss: but, to take
a single example from morphology, we begin to understand the
possibihty, and to comprehend the probable meaning, of the all but
sudden appearance on the earth of such exaggerated and almost
monstrous forms as those of the great secondary reptiles and the

500

Pure Shetland

10 20

Age, in months

30

40

Fig. 83.

Effect of cross-breeding on rate of growth in Shetland ponies.
From Walton and Hammond's data.*

great tertiary mammals f. We begin to see that it is in order to
account not for the appearance but for the disappearance of such
forms as these that natural selection must be invoked. And we
then, I think, draw near to the conclusion that what is true of these
is universally true, and that the great function of natural selection

* Walton and Hammond, Proc. R.S. (B), No. 840, p. 317, 1938.
t Cf. also Dendy, Evolutionary Biology, 1912, p. 408.

270 THE RATE OF GROWTH [ch.

is not to originate* but to remove: donee ad interitum genus id
natura redegitf.

The world of things living, like the world of things inanimate,
grows of itself, and pursues its ceaseless course of creative evolution.
It has room, wide but not unbounded, for variety of living form
and structure, as these tend towards their seemingly endless but
yet strictly hmited possibilities of permutation and degree^ it has
room for the great, and for the small, room for the weak and for the
strong. Environment and circumstance do not always make a
prison, wherein perforce the organism must either live or die; for
the ways of hfe may be changed, and many a refuge found, before
the sentence of unfitness is pronounced and the penalty of exter-
mination paid. But there comes a time when "variation," in form,
dimensions, or other qualities of the organism, goes further than is
compatible with all the means at hand of health and welfare for
the individual and the stock; when, under the active and creative
stimulus of forces from within and from without, the active and
creative energies of growth pass the bounds of physical and
physiological equilibrium: and so reach the Hmits which, as again
Lucretius tells us, natural law has set between what may and what
may not be,

et quid quaeque queant per foedera natural
quid porro nequeant.

Then, at last, we are entitled to use the customary metaphor, and
to see in natural selection an inexorable force whose function is not
to create but to destroy — to weed, to prune, to cut down and to
cast into the fire J.

* So said Yves Delage {UherediU, 1903, p. 397): "La selection naturelle est un
principe admirable et parfaitement juste. Tout le monde est d'accord sur ce point.
Mais ou Ton n'est pas d'accord, c'est sur la limite de sa puissance et sur la question
de savoir si elle pent engendrer des formes specifiques nouvelles. II semble bien
demontre aujourd'hui qu'elle ne le pent pas.''

t Lucret. v, 875, "Lucretius nowhere seems to recognise the possibility of
improvement or change of species by 'natural selection'; the animals remain as
they were at the first, except that the weaker and more useless kinds have been
crushed out. Hence he stands in marked contrast with modern evolutionists."
Kelsey's note, ad loc.

X Even after we have so narrowed its scope and sphere, natural selection is
still a hard saying ; for the causes of extinction are wellnigh as hard to understand
as are those of the origin of species. If we assert (as has been lightly and too

Ill] OF REGENERATIVE GROWTH 271

Of regeneration, or growth and repair

The phenomenon of regeneration, or the restoration of lost or
amputated parts, is a particular case of growth which deserves
separate consideration. It is a property manifested in a high
degree among invertebrates and many cold-blooded vertebrates,
diminishing as we ascend the scale, until it lessens down in the
warm-blooded animals to that vis smedicatrix which heals a wound.
Ever since the days of Aristotle, and still more since the experiments
of Trembley, Reaumur and Spallanzani in the eighteenth century,
physiologist and psychologist alike have recognised that the pheno-
menon is both perplexing and important. "Its discovery," said
Spallanzani, " was an immense addition to the riches of organic philo-
sophy, and an inexhaustible source of meditation for the philosopher."
The general phenomenon is amply treated of elsewhere*, and we
need only deal with it in its immediate relation to growth.

Regeneration, like growth in other cases, proceeds with a velocity
which varies according to a definite law; the rate varies with the
time, and we may study it as velocity and as acceleration. Let us
take, as an instance, Miss M. L. Durbin's measurements of the rate
.of regeneration of tadpoles' tails : the rate being measured in terms
of length, or longitudinal increment f. From a number of tadpoles,
whose average length was in one experiment 34 mm., and in another
49 mm., about half the tail was cut off, and the average amounts
regenerated in successive periods are shewn as follows :

Days 3 5 7 10 12 14 17 18 24 28 30

Amount regenerated (mm.):

First experiment 1-4 — 3-4 4-3 — 5-2 — 5-5 6-2 — 6o

Second „ 0-9 2-2 3-7 5-2 60 6-4 7-1 — 7-6 8-2 8-4

confidently done) that Smilodon perished on account of its gigantic tusks, that
Teleosaurus was handicapped by its exaggerated snout, or Stegosaurus weighed
down by its intolerable load of armour, we may call to mind kindred forms where
similar conditions did not lead to rapid extermination, or where extinction ensued
apart from any such apparent and visible disadvantages. Cf. F. A. Lucas, On
momentum in variation, Amer. Nat. xli, p. 46, 1907.

* See Professor T. H. Morgan's Regeneration (316 pp.), 1901, for a full account
and copious bibliography. The early experiments on regeneration, by V^allisneri,
Dicquemare, Spallanzani, Reaumur, Trembley, Baster, Bonnet and others, are
epitomised by Haller, Elementa Physiologiae, viii, pp. 156 seq.

t Journ. Exper. Zool. vii, p. 397, 1909.

272

THE RATE OF GROWTH

[CH.

Both experiments give us fairly smooth curves of growth within
the period of the observations; and, with a shght and easy extra-
polation, both curves draw to the base-hne at zero (Fig. 84). More-

Fig. 84. Curve of regenerative growth in tadpoles' tails.
From M. L. Durbin's data.

Fig. 85. Tadpoles' tails : , amount regenerated daily, in mm.
(Smoothed curve).

over, if from the smoothed curves we deduce the daily increments,
we get (Fig. 85) a bell-shaped curve similar to (or to all appearance
identical with) a skew curve of error. In point of fact, this instance
of regeneration is a very ordinary example of growth, with its

Ill]

OF REGENERATION

273

S-shaped curve of integration and its bell-shaped differential curve,
just as we have seen it in simple cases, or simple phases, of 'the
growth of a population or an individual.

If we amputate one limb of a pair in some animal with rapid
powers of regeneration, we may compare from time to time the
dimensions of the regenerating hmb with those of its uninjured
fellow, and so deal with a relative rather than an absolute velocity.
The legs of insect-larvae are easily restored, but after pupation no
further growth or regeneration takes place. An easy experiment,
then, is to remove a limb in larvae of various ages, and to compare

100 120

Fig. 86. Regenerative growth in mealworms' legs.

at leisure in the pupa the dimensions of the new Hmb with the old.
The following much-abbreviated table shews the gradual increase
of a regenerating limb in a mealworm, up to final equahty with the
normal Hmb, the rate varying according to the usual S-shaped
curve* (Fig. 86).

Rate of regeneration in the mealworm (Tenebrio moHtor, larva)

Days after amputation
% ratio of new limb to old

16 21 25 34 44 58 70 100 121
7 11 20 29 42 71 83 91 100

* From J. Krizenecky, Versuch zur statisch-graphischen Untersuchung . . .der
Regenerationsvorgange, Arch. f. Entw. Mech. xxxix, 1914; xlii, 1917.

274 THE RATE OF GROWTH [ch.

Some writers have found the curve of regenerative growth to be
different from the curve of ordinary growth, and have commented
on the apparent difference; but they have been misled (as it seems
to me) by the fact that regeneration is seen from the start or very
nearly so, while the ordinary curves of growth, as they are usually
presented to us, date not from the beginning of growth, but from
the comparatively late, and unimportant, and even fallacious epoch
of birth. A complete curve of growth, starting from zero, has the
same essential characteristics as the regeneration curve.

Indeed the more we consider the phenomenon of regeneration,
the more plainly does it shew itself to us as but a particular case
of the general phenomenon of growth*, following the same lines,
obeying the same laws, and merely started into activity by the
special stimulus, direct or indirect, caused by the infliction of a
wound. Neither more nor less than in other problems of physiology
are we called upon, in the case of regeneration, to indulge in
metaphysical speculation, or to dwell upon the beneficent purpose
which seemingly underhes this process of heahng and repair.

It is a very general rule, though not a universal one, that
regeneration tends to fall somewhat short of a complete restoration
of the lost part; a certain percentage only of the lost tissues is
restored. This fact was well known to some of those old .investi-
gators, who, like the Abbe Trembley and hke Voltaire, found a
fascination in the study of artificial injury and the regeneration
which followed it. Sir John Graham Dalyell, for instance, says, in
the course of an admirable paragraph on regeneration!: "The
reproductive faculty ... is not confined to one pjortion, but may
extend over many; and it may ensue even in relation to the
regenerated portion more than once. Nevertheless, the faculty
gradually weakens, so that in general every successive regeneration
is smaller and more imperfect than the organisation preceding it;
and at length it is exhausted."

* The experiments of Loeb on the growth of Tubularia in various saline
solutions, referred to on p. 24.5, might as well or better have been referred to under
the heading of regeneration, as they were performed on cut pieces of the zoophyte.
(Cf. Morgan, op. cit. p. 35.)

t Powers of the Creator, i, p. 7, 1851. See also Rare and Remarkable Animals,
u, pp. 17-19, 90, 1847.

Ill] OF REGENERATION 275

In certain minute animals, such as the Infusoria, in which the
capacity for regeneration is so great that the entire animal may-
be restored from a mere fragment, it becomes of great interest to
discover whether there be some definite size at which the fragment
ceases to display this power. This question has been studied by
Lillie*, who found that in Stentor, while still smaller fragments were
capable of surviving for days, the smallest portions capable of
regeneration were of a size equal to a sphere of about 80 /x in
diameter, that is to say of a volume equal to about one twenty-
seventh of the average entire animal. He arrives at the remarkable
conclusion that for this, and for all other species of animals, there
is a "minimal organisation mass," that is to say a "minimal mass
of definite size consisting of nucleus and cytoplasm within which
the organisation of the species can just find its latent expression."
And in like manner, Boverif has shewn that the fragment of a sea-
urchin's egg capable of growing up into a new embryo, and so
discharging the complete functions of an entire and uninjured ovum,
reaches its limit at about one-twentieth of the original egg — other
writers having found a Hmit at about one-fourth. These magnitudes,
small as they are, represent objects easily visible under a low power
of the microscope, and so stand in a very different category to the
minimal magnitudes in which fife itself can be manifested, and
which we have discussed in another chapter.

The Bermuda "hfe-plant" (Bryophyllum calycinum) has so
remarkable a power of regeneration that a single leaf, kept damp,
sprouts into fresh leaves and rootlets which only need nourishment
to grow into a new plant. If a stem bearing two opposite leaves
be split asunder, the two co-equal sister-leaves will produce (as we
might indeed expect) equal masses of shoots in equal times, whether
these shoots be many or fe^^; and, if one leaf of the pair have part
cut off it and the other be left intact, the amount of new growth

* F. R. Lillie, The smallest parts of Stentor capable of regeneration, Journ.
Morphology, xii, p. 239, 1897.

t -Boveri, Entwicklungsfahigkeit kernloser Seeigeleier, etc.. Arch. /. Entw. Mech.
u, 1895. See also Morgan, Studies of the partial larvae of Sphaerechinus, ibid.
1895; J. Loeb, On the limits of divisibility of living matter, Biol. Lectures, 1894;
Pfluger's Archiv, lix, 1894, etc. Bonnet studied the same problem a hundred
and seventy years ago, and found that the smallest part of the worm Lumbriculus
capable of regenerating was 1^ lines (3-4 mm.) long. For other references and
discussion see H. Przibram, Form und Formel, 1922, ch. v.

276

THE RATE OF GROWTH

[CH.

will be in direct and precise proportion to the mass of the leaf from
which it grew. The leaf is all the while a living tissue, manu-
facturing material to build its own offshoots ; and we have a simple
case of the law of mass action in the relation between the mass of
the leaf with its 'included chlorophyll and. that of its regenerated
oifshoot*.

16 18 20
days

Fig. 87. Relation between the percentage amount of tail removed, the percentage
restored, and the time required for its restoration. Constructed from M. M.
Ellis's data.

A number of phenomena connected with the linear rate of
regeneration are illustrated and epitomised in the accompanying
diagram (Fig. 87), which I have constructed from certain data
given by Ellis in a paper on the relation of the amount of tail
regenerated to the amount removed, in tadpoles. These data are
summarised in the next table. The tadpoles were all very much

* Jacques Loeb, The law controlling the quantity and rate of regeneration,
Proc. Nat. Acad. Sci. iv, pp. 117-121, 1918; Journ. Gen. Physiol, i, pp. 81-96,
1918; Botan. Oaz. lxv, pp. 150-174, 1918.

Ill] OF REGENERATION 277

of a size, about 40 mm. ; the average length of tail was very near
to 26 mm., or 65 per cent, of the whole body-length; and in four
series of experiments about 10, 20, 40 and 60 per cent, of the tail
were severally removed. The amount regenerated in successive
intervals of three days is shewn in our table. By plotting the
actual amounts regenerated against these three-day intervals of
time, we may interpolate values for the time taken to regenerate
definite percentage amounts, 5 per cent., 10 per cent., etc. of the
amount removed; and my diagram is constructed from the four
sets of values thus obtained, that is to say from the four sets of
experiments which differed from one another in the amount of tail
amputated. To these we have to add the general result of a fifth
series of experiments, which shewed that when as much as 75 per
cent, of the tail was cut off, no regeneration took place at all, but
the animal presently died. In our diagram, then, each curve
indicates the time taken to regenerate n per cent, of the amount
removed. All the curves converge towards infinity of time, when
the amount removed approaches 75 per cent, of the whole; and all
start from zero, for nothing is regenerated where nothing had been
destroyed.

The rate of regenerative growth in tadpoles' tails
(After M. M. Ellis, Journ. Exp. Zool. vii, ^.421, 1909)

Body

Tail

Amount Per cent.

/o

amount regenerated

in days

length

length

removed

of tail

r

— ^

Series*

mm.

mm.

mm.

removed

3

6

9

12

15

18

32

39-575

25-895

3-2

12-36

13

31

44

44

44

44

44

P

40-21

26-13

5-28

20-20

10

29

40

44

44

44

44

R

39-86

25-70

10-4

40-50

6

20

31

40

48

48

48

S

40-34

2611

14-8

56-7

16

33

39

45

48

48

* Each series gives the mean of 20 experiments.

The amount regenerated varies also with the age of the tadpole,
and with other factors such as temperature; in short, for any given
age or size of tadpole, and for various temperatures, and doubtless
for other varying physical conditions, a similar diagram might be
constructed!.

The power of reproducing, or regenerating, a lost limb is par-

t Cf. also C. Zeleny, Factors controlling the rate of regeneration, Illinois Biol-
Monographs, iii, p. 1, 1916.

278 THE RATE OF GROWTH [ch.

ticularly well developed in arthropod animals, and is sometimes
accompanied by remarkable modification of the form of the
regenerated limb. A case in point, which has attracted much
attention, occurs in connection with the claws of certain Crustacea*.

In many of these we have an asymmetry of the great claws,
one being larger than the other and also more or less different in
form. For instance in the common lobster, one claw, the larger
of the two, is provided with a few great "crushing" teeth, while
the smaller claw has more numerous teeth, small and serrated.
Though Aristotle thought otherwise, it appears that the crushing-
claw may be on the right' or left side, indifferently ; whether it be
on one or the other is a matter of "chance." It is otherwise in
many other Crustacea, where the larger and more powerful claw is
always left or right, as the case may be, according to the species:
where, in other words, the "probability" of the large or the small
claw being left or being right is tantamount to certainty f.

As we have already seen, the one claw is the larger because it
has grown the faster; it has a higher "coefficient of growth," and
accordingly, as age advances, the disproportion between the two
claws becomes more and more evident. Moreover, we must assume
that the characteristic form of the claw is a "function" of its
magnitude ; the knobbiness is a phenomenon coincident with
growth, and we never, under any circumstances, find the smaller
claw with big crushing teeth and the big claw with Httle serrate
ones. There are many other somewhat similar cases where size
and form are manifestly correlated, and we have already seen, to
some extent, how the phenomenon of growth is often accompanied
by such ratios of velocity as lead inevitably to changes of form.
Meanwhile, then, we must simply assume that the essential difference
between the two claws is one of magnitude, with which a certain
differentiation of form is inseparably associated.

* Cf. H. Przibram, Scheerenumkehr bei dekapoden Crustaceen, Arch. f. Entw.
Mech. XIX, pp. 181-247, 1905; xxv, pp. 266-344, 1907; Emmel, ibid, xxii, p. 542,
1906; Regeneration of lost parts in lobster, Bep. Comm. Inland Fisheries, Rhode
Island, XXXV, xxxvi, 1905-6; Science (N.S.), xxvi, pp. 83-87, 1907; Zeleny,
Compensatory regulation, Journ. Exp. Zool. ii, pp. 1-102, 347-369, 1905; etc.

t Lobsters are occasionally found with two symmetrical claws: which are then
usually serrated, sometimes (but very rarely) both blunt-toothed. Cf. W. T. Caiman,
P.Z.8. 1906, pp. 633, 634, and reff.

Ill] OF REGENERATION 279

If we amputate a claw, or if, as often happens, the crab "casts
it off," it undergoes a process of regeneration — it grows anew,
and does so with an accelerated velocity which ceases when
equilibrium of the parts is once more attained : the accelerated velocity
being a case in point to illustrate that vis revulsionis of Haller to
which we have already referred.

With the help of this principle, Przibram accounts for certain
curious phenomena which accompany the process of regeneration.
As his experiments and those of Morgan shew, if the large or knobby
claw (A) be removed, there are certain cases, e.g. the common
lobster, where it is directly regenerated. In other cases, e.g.
Alpheus*, the other claw (B) assumes the size and form of that
which was amputated, while the latter regenerates itself in the
form of the lesser and weaker one; A and B have apparently
changed places. In a third case, as in the hermit-crabs, the A-
claw regenerates itself as a small or 5-claw, but the 5-claw
remains for a time unaltered, though slowly and in the course of
repeated moults it later on assumes the large and heavily toothed
^-form.

Much has been written on this phenomenon, but in essence it is
very simple. It depends upon the respective rates of growth, upon
a ratio between the rate of regeneration and the rate of growth of
the uninjured limb: that is to say, on the familiar phenomenon of
unequal growth, or, as it has been called, heterogony*. It is com-
plicated a little, however, by the possibility of the uninjured limb
growing all the faster for a time after the animal has been relieved
of the other. From the time of amputation, say of A, A begins to
grow from zero, with a high "regenerative" velocity; while B,
starting from a definite magnitude, continues to increase with its
normal or perhaps somewhat accelerated velocity. The ratio
between the two velocities of growth will determine whether, by a
given time, A has equalled, outstripped, or still fallen short of the
magnitude of B.

That this is the gist of the whole problem is confirmed (if con-
firmation be necessary) by certain experiments of Wilson's. It is

* E.' B. Wilson, Reversal of symmetry in Alpheus heterocheles, Biol. Bull, iv,
p. 197, 1903.
t See p. 205.

280 THE RATE OF GROWTH [ch.

known that by section of the nerve to a crab's claw, its growth is
retarded, and as the general growth of the animal proceeds the claw
comes to appear stunted or dwarfed. Now in such a case as that
of Alpheus, we have seen that the rate of regenerative growth in an
amputated large claw fails to let it reach or overtake the magnitude
of the growing little claw: which latter, in short, now appears as
the big one. But if at the same time as we amputate the big claw
we also sever the nerve to the lesser one, we so far slow down the
latter's growth that the other is able to make up to it, and in this
case the two claws continue to grow at approximately equal rates,
or in other words continue of coequal size.

The phenomenon of regeneration goes some little way towards
helping us to comprehend the phenomenon of "multiplication by
fission," as it is exemplified in its simpler cases in many worms and
worm-like animals. For physical reasons which we shall have to
study in another chapter, there is a natural tendency for any tube,
if it have the properties of a fluid or semi-fluid substance, to break
up into segments after it comes to a certain length*; and nothing
can prevent its doing so except the presence of some controlling
force, such for instance as may be due to the pressure of some
external support, or some superficial thickening or other intrinsic
rigidity of its own substance. If we add to this natural tendency
towards fission of a cylindrical or tubular worm, the ordinary
phenomenon of regeneration, we have all that is essentially implied
in "reproduction by fission." And in so far as the process rests
upon a physical principle, or natural tendency, we may account for
its occurrence in a great variety of animals, zoologically dissimilar;
and for its presence here and absence there, in forms which are
materially different in a physical sense, though zoologically speaking
they are very closely allied.

But the phenomena of regeneration, like all the other phenomena
of growth, soon carry us far afield, and we must draw this long
discussion to a close.

* A morphological polarity, or essential difference between one end and the other
of a segment, is important even in so simple a case as the internode of a hydroid
zoophyte; and an electrical polarity seems always to accompany it. Cf. A. P.
Matthews, Amer. Journ. Physiology, viii, j). 294, 1903; E. J. Lund, Journ. Exper.
Zool. xxxiy, pp. 477-493; xxxvi, pp. 477^94, 1921-22.

Ill] THE RATE OF GROWTH 281

Summary and Conclusion

For the main features which appear to be common to all curves
of growth we may hope to have, some day, a simple explanation.
In particular we should like to know the plain meaning of that point
of inflection, or abrupt change from an increasing to a decreasing
velocity of growth, which all our curves, and especially our accelera-
tion curves, demonstrate the existence of, provided only that they
include the initial- stages of the whole phenomenon: just as we
should also hke to have a full physical or physiological explanation
of the gradually diminishing velocity of growth which follows, and
which (though subject to temporary interruption or abeyance) is
on the whole characteristic of growth in all cases whatsoever. In
short, the characteristic form of the curve of growth in length (or
any other linear dimension) is a phenomenon which we are at
present little able to explain, but which presents us with a definite
and attractive problem for future solution. It would look as
though the abrupt change in velocity must be due, either to a change
in that pressure outwards from within by which the "forces of
growth" make themselves manifest, or to a change in the resistances
against which they act, that is to say the tension of the surface;
and this latter force we do not by any means limit to "surface-
tension" proper, but may extend to the development of a more or
less resistant membrane or "skin," or even to the resistance of fibres
or other histological elements binding the boundary layers to the
parts within*. I take it that the sudden arrest of velocity is much
more likely to be due to a sudden increase of resistance than to a
sudden diminution of internal energies: in other words, I suspect
that it is coincident with some notable event of histological
differentiation, such as the rapid formation of a comparatively firm
skin ; and that the dwindling of velocities, or the negative accelera-
tion, which follows, is the resultant or composite effect of waning
forces of growth on the one hand, and increasing superficial resistance

* It is natural to suppose the cell-wall less rigid, or more plastic, in the growing
tissue than in the full-grown or resting cell. It has been suggested that this plasticity-
is due to, or is increased by, auxins, whether in the course of nature, or in our
stimulation of growth by the use of these bodies. Cf. H. Soding, Jahrb. d. wiss. Bot.
Lxxiv, p. 127; 1931.

282 THE RATE OF GROWTH [ch.

on the other. This is as much as to say that growth, while its own
energy tends to increase, leads also, after a while, to the establish-
ment of resistances which check its own further increase.

Our knowledge of the whole complex phenomenon of growth is
so scanty that it may seem rash to advance even this tentative
suggestion. But yet there are one or two known facts which seem
to bear upon the question, and to indicate at least the manner in
which a varying resistance to expansion may affect the velocity
of growth. For instance, it has been shewn by Frazee* that
electrical stimulation of tadpoles, with small current density and
low voltage, increases the rate of regenerative growth. As just
such an electrification would tend to lower the surface-tension, and
accordingly decrease the external resistance, the experiment would
seem to support, in some slight degree, the suggestion which I have
made.

To another important aspect of regeneration we can do no more
than allude. The Planarian worms rival Hydra itself in their powers
of regeneration; and in both cases even small bits of the animal
are likely to include endoderm cells capable of intracellular digestion,
whereby the fragment is enabled to live and to grow. Now if a
Planarian worm be cut in separate pieces and these be suffered to
grow and regenerate, they do so in a definite and orderly way ; that
part of a shce or fragment which had been nearer tq the original
head will develop a head, and a tail will be regenerated at the
opposite end of the same fragment, the end which Had been tailward
in the beginning; the amputated fragments possess sides and ends,
a front end and a hind end, like the entire worm; in short, they
retain their polarity. This remarkable discovery is due to Child,
who has amplified and extended it in various instructive ways.
The existence of two poles, positive and negative, implies a
"gradient" between them. It means that one part leads and
another follows; that one part is dominant, or prepotent over the
rest, whether in regenerative growth or embryonic development.

We may summarise, as follows, the main results of the foregoing
discussion :

(1) Except in certain minute organisms, whose form (hke that

* Journ. Ezper. ZooL vii, p. 457, 1909.

Ill] SUMMARY AND CONCLUSION 283

of a drop of water) is due to the direct action of the molecular forces,
we may look upon the form of an organism as a " function of growth,"
or a direct consequence of growth whose rate varies in its different
directions. In a newer language we might call the form of an
organism an "event in space-time," and not merely a "configuration
in space."

(2) Growth varies in rate in an orderly way, or is subject, like
other physiological activities, to definite "laws." The rates differ
in degree, or form "gradients," from one point of an organism to
another; the rates in different parts and in different directions
tend to maintain more or less constant ratios to one another in
each organism ; , and to the regularity and constancy of these relative
rates of growth is due the fact that the form of the organism is in
general regular and constant.

(3) Nevertheless, the ratio of velocities in different directions is
not absolutely constant, but tends to alter in course of time, or to
fluctuate in an orderly way; and to these progressive changes are
due the changes of form which accompany development, and the
slower changes which continue perceptibly in after hfe.

(4) Rate of growth depends on the age of the organism. It has
a maximum somewhat early in hfe, after which epoch of maximum
it slowly declines.

(5) Rate of growth is directly affected by temperature, and by
other physical conditions: the influence of temperature being
notably large in the case of cold-blooded or " poecilothermic "
animals. Growth tends in these latter to be asymptotic, becoming
slower but never ending with old age.

(6) It is markedly affected, in the way of acceleration or retarda-
tion, at certain physiological epochs of hfe, such as birth, puberty
or metamorphosis.

(7) Under certain circumstances, growth may be negative, the
organism growing smaller; and such negative growth is a common
accompaniment of metamorphosis, and a frequent concomitant of
old age.

(8) The phenomenon of regeneration is associated with a large
transitory increase in the rate of growth (or acceleration of growth)
in the region of injury; in other respects regenerative growth is
similar to ordinary growth in all its essential phenomena.

284 THE RATE OF GROWTH [ch.

In this discussion of growth, we have left out of account a vast
number of processes or phenomena in the physiological mechanism
of the body, by which growth is effected and controlled. We have
dealt with growth in its relation to magnitude, and to that relativity
of magnitudes which constitutes form; and so we have studied it
as a phenomenon which stands at the beginning of a morphological,
rather than at the end of a physiological enquiry. Under these
restrictions, we have treated it as far as possible, or in such fashion
as our present knowledge permits, on strictly physical lines. That
is to say, we rule "heredity" or any such concept out of our present
account, however true, however important, however indispen-
sable in another setting of the story, such a concept may be.
In physics "on admet que I'etat actuel du monde ne depend que du
passe le. plus proche, sans etre influence, pour ainsi dire, par le
souvenir d'un passe lointain*." This is the concept to which the
differential equation gives expression; it is the step which Newton
took when he left Kepler behind.

In all its aspects, and not least in its relation to form, the growth
of organisms has many analogies, some close, some more remote,
among inanimate things. As the waves grow when the winds strive
with the other forces which govern the movements of the surface
of the sea, as the heap grows when we pour corn out of a sack, as
the crystal grows when from the surrounding solution the proper
molecules fall into their appropriate places: so in all these cases,
very much as in the organism itself, is growth accompanied by
change of form, and by a development of definite shapes and
contours. And in these cases (as in all other mechanical phenomena),
we are led to equate our various magnitudes with time, and so to
recognise that growth is essentially a question of rate, or of velocity.

The diiferences of form, and changes of form, which are brought
about by varying rates (or "laws") of growth, are essentially the
same phenomenon whether they be episodes in the life-history of
the individual, or manifest themselves as the distinctive charac-
teristics of what we call separate species of the race. From one
form, or one ratio of magnitude, to another there is but one straight
and direct road of transformation, be the journey taken fast or

* Cf. H. Poincare, La physique generale et la physique mathematique, Rev.
gin. des Sciences, xi, p. 1167, 1900.

Ill] SUMMARY AND CONCLUSION 285

slow; and if the transformation take place at all, it will in all
likelihood proceed in the self-same way, whether it occur within
the Ufetime of an individual or during the long ancestral history of
a race. No small part of what is known as Wolff's or von Baer*s
law, that the individual organism tends to pass through the phases
characteristic of its ancestors, or that the life-history of the individual
tends to recapitulate the ancestral history of its race, lies wrapped
up in this simple account of the relation between growth and form.
But enough of this discussion. Let us leave for a while the
subject of the growth of the organism, and attempt to study the
conformation, within and without, of the individual cell.

CHAPTER IV

ON THE INTERNAL FORM AND STRUCTURE
OF THE CELL

In the early days of the cell-theory, a hundred years ago, Goodsir
was wont to speak of cells as "centres of growth" or "centres of
nutrition," and to consider them as essentially "centres of force*".
He looked forward to a time when the forces connected with the
cell should be particularly investigated : when, that is to say, minute
anatomy should be studied in its dynamical aspect. "When this
branch of enquiry," he says, "shall have been opened up, we shall
expect to have a science of organic forces, having direct relation
to anatomy, the science of organic forms." And likewise, long
afterwards, Giard contemplated a science of morphodynatnique — but
still looked upon it as forming so guarded and hidden a "territoire
scientifique, que la plupart des naturalistes de nos jours ne le verront
que comme Moise vit la terre promise, seulement de loin et sans
pouvoir y entrerf ."

To the external forms of cells, and to the forces which produce
and modify these forms, we shall pay attention in a later chapter.
But there are forms and configurations of matter within the cell
which also deserve to be studied with due regard to the forces,
known or unknown, of whose resultant they are the visible
expression.

* Anatomical and Pathological Observations, p. 3, 1845; Anatomical Memoirs,
II, p. 392, 1868. This was a notable improvement on the "kleine wirkungsfahige
Zentren oder Elementen" of the Cellularpathologie. Goodsir seems to have been
seeking an analogy between the living cell and the physical atom, which Faraday,
following Boscovich, had been speaking of as a centre of force in the very year
before Goodsir published his Observations: see Faraday's Speculations concerning
Electrical Conductivity and the Nature of Matter, 1844. For Newton's "molecules"
had been turned by his successors into material points; and it was Boscovich (in
1758) who first regarded these material points as mere persistent centres of force.
It was the same fertile conception of a centre of force which led Rutherford, later
on, to the discovery of the nucleus of the atom.

t A. Giard, L'oeuf et les debuts de revolution, Bull. Sci. du Nord de la Fr. vm,
pp. 252-258, 1876.

CH. IV] THE CELL THEORY . 287

In the long interval since Goodsir's day, the visible structure,
the conformation and configuration, of the cell, has been studied
far more abundantly than the purely dynamic problems which are
associated therewith. The overwhelming progress of microscopic
observation has multipHed our knowledge of cellular and intra-
cellular structure ; and to the multitude of visible structures it has
been often easier to attribute virtues than to ascribe intelUgible
functions or modes of action. But here and there nevertheless,
throughout the whole hteratiire of the subject, we find recognition
of the inevitable fact that dynamical problems lie behind the
morphological problems of the cell.

Biitschli pointed out sixty years ago, with emphatic clearness,
the failure of morphological methods and the need for physical
methods if we were to penetrate deeper into the essential nature of
the cell*. And such men as Loeb and Whitman, Driesch and Roux,
and not a few besides, have pursued the same train of thought and
similar methods of enquiry.

Whitman t, for instance, puts the case in a nutshell when, in
speaking of the so-called " caryokinetic " phenomena of nuclear
division, he reminds us that the leading idea in the term ''caryo-
kinesis'' is ynotion — "motion viewed as an exponent of forces
residing in, or acting upon, the nucleus. It regards the nucleus
as a seat of energy, which displays itself in phenomena of motion X-^'

In short it would seem evident that, except in relation to a
dynamical investigation, the mere study of cell structure has but

* Entwickelungsvorgdnge der Eizelle, 1876; Investigations on Microscopic Foams
and Protoplasm, p. 1, 1894.

t Journ. Morphology, i, p. 229, 1887.

t While it has been very common to look upon the phenomena of mitosis as
sufficiently explained by the results towards which they seem to lead, we may find
here and there a strong protest against this mode of interpretation. The following
is a case in point: ''On a tente d'etablir dans la mitose dite primitive plusieurs
categories, plusieurs types de mitose. On a choisi le plus souvent comme base
de ces systemes des concepts abstraits et teleologiques : repartition plus ou moins
exacte de la chromatine entre les deux noyaux-fils suivant qu'il y a ou non des
chromosomes (Dangeard), distribution particuliere et signification dualiste des
substances nucleaires (substance kinetique et substance generative ou hereditaire,
Ilartmann et ses eleves), etc. Pour moi tous ces essais sont a rejeter categorique-
ment a cause de leur caractere finaliste; de plus, ils sont construits sur des concepts
non demontres, et qui parfois representent des generalisations absolument erronees.''
A. Alexeieflf, Archiv fiir Protistenkunde, xix, p. 344, 1913.

288 ON THE INTERNAL FORM [ch.

little value of its own. That a given cell, an ovum for instance,
contains this or that visible substance or structure, germinal vesicle
or germinal spot, chromatin or achromatin, chromosomes or centro-
somes, obviously gives no explanation of the activities of the cell.
And in all such hypotheses as that of "pangenesis," in all the
theories which attribute specific properties to micellae, chromosomes,
idioplasts, ids, or other constituent particles of protoplasm or of
the cell, we are apt to fall into the error of attributing to matter
what is due to energy and is manifested in force: or, more strictly
speaking, of attributing to material particles individually what is
due to the energy of their collocation.

The tendency is a very natural one, as knowledge of structure
increases, to ascribe particular virtues to the material structures
themselves, and the error is one into which the disciple is Ukely
to fall but "of which we need not suspect the master-mind. The
dynamical aspect of the case was in all probability kept well in view
by those who, Hke Goodsir himself, first attacked the problem of
the cell and originated our conceptions of its nature and functions*.

If we speak, as Weismann and others speak, of an "hereditary
sitbstance,'' a substance which is spHt off from the parent-body, and
which hands on to the new generation the characteristics of the old,
we can only justify our mode of speech by the assumption that that
particular portion of matter is the essential vehicle of a particular
charge or distribution of energy, in which is involved the capabihty
of producing motion, or of doing "work." For, as Newton said,
to tell us that a thing "is endowed with an occult specific quahtyf,
by which it acts and produces manifest effects, is to tell us nothing ;
but to derive two or three general principles of motion { from

* See also {int. al.) R. S. Lillie's papers on the physiology of cell-division in the
Journ. Exper. Physiology; especially No. vi, Rhythmical changes in the resistance
of the dividing sea-urchin egg, ibid, xvi, pp. 369-402, 1916.

f Such as the vertu donnitive which accounts for the soporific action of opium.
We are now more apt. as Le Dantec says, to substitute for this occult quality the
hypothetical substance dormitin.

X This is the old philosophic axiom writ large: Ignorato motu, ignoratur -natura;
which again is but an adaptation of Aristotle's phrase, 17 dpx^ ^V^ Kivrjcrecxjs, as
equivalent to the "Efiicient Cause." FitzGerald holds that "all explanation
consists in a description of underlying motions" {Scientific Writings, 1902, p. 385);
and Oliver Lodge remarked, "You can move Matter; it is the only thing you can
do to it."

IV] AND STRUCTURE OF THE CELL 289

phenomena would be a very great step in philosophy, though the
causes of those principles were not yet discovered." The things
which we see in the cell are less important than the actions which
we recognise in the cell; and these latter we must especially,
scrutinise, in the hope of discovering how far they may be attributed
to the simple and well-known physical forces, and how far they be
relevant or irrelevant to the phenomena which we associate with,
and deem essential to, the manifestation of life. It may be that in
this way we shall in time draw nigh to the recognition of a specific
and ultimate residuum.

And lacking, as we still do lack, direct knowledge of the
actual forces inherent in the cell, we may yet learn something
of their distribution, if not also of their nature, from the
outward and inward configuration of the cell and from the
changes taking place in this configuration; that is to say from
the movements of matter, the kinetic phenomena, which the forces
in action set up.

The fact that the germ-cell develops into a very complex structure
is no absolute proof that the cell itself is structurally a very com-
phcated mechanism : nor yet does it prove, though this is somewhat
less obvious, that the forces at work or latent within it are especially
numerous and complex. If we blow into a bowl of soapsuds and
raise a great mass of many-hued and variously shaped bubbles, if
we explode a rocket and watch the regular and beautiful configura-
tion of its falhng streamers, if we consider the wonders of a Hmestone
cavern which a filtering stream has filled with stalactites, we soon
perceive that in all these cases we have begun with an initial system
of very slight complexity, whose structure in no way foreshadowed
the result, and whose comparatively simple intrinsic forces only
play their part by complex interaction with the equally simple
forces of the surrounding medium. In an earlier age, men sought
for the visible embryo, even for the homunculus, within the repro-
ductive cells; and to this day we scrutinise these cells for visible
structure, unable to free ourselves from that old doctrine of
' ' pre-f ormation * . "

Moreover, the microscope seemed to substantiate the idea (which

* As when Nageli concluded that the organism is, in a certain sense, "vorge-
bildet"; Beitr. zur wiss. Botanik, ii, 1860.

290 ON THE INTERNAL FORM [ch.

we may trace back to Leibniz* and to Hobbest), that there is no
Hmit to the mechanical complexity which we may postulate in an
organism, and no limit, therefore, to the hypotheses which we may
rest thereon. But no microscopical examination of a stick of sealing-
wax, no study of the material of which it is composed, can enlighten
us as to its electrical manifestations or properties. Matter of itself
has no power to do, to make, or to become: it is in energy that
all these potentiaUties reside, energy invisibly associated with the
material system, and in interaction with the energies of the
surrounding universe.

That ''function presupposes structure" has been declared an
accepted axiom of biology. Who it was that so formula ted the
aphorism I do not know; but as regards the structure of the cell
it harks back to Briicke, with whose demand for a mechanism, or
an organisation, within the cell histologists have ever since been
trying to comply J. But unless we mean to include thereby
invisible, and merely chemicu,l or molecular, structure, we come at
once on dangerous ground. For we have seen in a former chapter
that organisms are known of magnitudes so nearly approaching the
molecalar, that everything which the morphologist is accustomed to
conceive as "structure" has become physically impossible; and
recent research tends to reduce, rather than to extend, our con-
ceptions of the visible structure necessarily inherent in living
protoplasm §. The microscopic structure which in the last resort

* "La matiere arrangee par une sagesse divine doit etre essentiellement organisee
partout. . .il y a machine dans les parties de la machine naturelle a I'infini." Siir le
principe de la Vie, p. 431 (Erdmann). This is the very converse of the doctrine
of the Atomists, who could not conceive a condition "w6/ dimidiae partis pars
semper hahehit Dimidiam partem, nee res praefiniet ulla.''

t Cf. an interesting passage from the Elements (i, p. 445, Molesworth's edit.),
quoted by Owen, Hunterian Lectures on the Invertebrates, 2nd ed. pp. 40, 41, 1855.

% "Wir miissen deshalb den lebenden Zellen, abgesehen von der Molekular-
structur der organischen Verbindungen welche sie enthalt, noch eine andere und
in anderer Weise complicirte Structur zuschreiben, und diese es ist welche wir
mit dem Namen Organisation bezeichnen," Briicke, Die Elementarorganismen,
Wiener Sitzungsber. xliv, 1861, p. 386; quoted by Wilson, The Cell, etc., p. 289.
Cf. also Hardy, Journ. Physiol, xxiv, 1899, p. 159.

§ The term protoplasm was first used by Purkinje, about 1839 or 1840 (cf.
Reichert, Arch. f. Anat. u. Physiol. 1841). But it was better defined and more
strictly used by Hugo von Mohl in his paper Ueber die Saftbewegung im Inneren
der Zellen, Botan. Zeitung, iv, col. 73-78, 89-94, 1846.

IV] AND STRUCTURE OF THE CELL 291

or in the simplest cases it seems to sheWj is that of a more or less
viscous colloid, or rather mixtm:e of colloids, and nothing more.
Now, as Clerk Maxwell puts it in discussing this very problem,
"one material system can differ from another only in the configura-
tion and motion which it has at a given instant*." If we cannot
assume differences in structure or configuration, we must assume
differences in motion, that is to say in energy. And if we cannot
do this, then indeed we are thrown back upon modes of reasoning
unauthorised in physical science, and shall find ourselves constrained
to assume, or to "admit, that the properties of a germ are not those
of a purely material system."

But we are by no means necessarily in this dilemma. For though
we come perilously near to it when we contemplate the lowest
orders of magnitude to which hfe has been attributed, yet in the
case of the ordinary cell, or ordinary egg or germ which is going
to develop into a complex organism, if we have no reason to assume
or to beheve that it comprises an intricate "mechanism," we mgiy
be quite sure, both on direct and indirect evidence, that, hke the
powder in our rocket, it is very heterogeneous in its structure.
It is a mixture of substances of various kinds, more or less fluid,
more or less mobile, influenced in various ways by chemical, electrical,
osmotic and other forces, and in their admixture separated by a
multitude of surfaces or boundaries, at which these or certain of
these forces are made manifest.

Indeed, such an arrangement as this is already enough to con-
stitute a "mechanism"; for we must be very careful not to let our
physical or physiological concept of mechanism be narrowed to an
interpretation of the term derived from the comphcated contrivances
of himaan skill. From the physical point of view, we understand
by a "mechanism" w^hatsoever checks or controls, and guides into
determinate paths, the workings of energy: in other words, what-
soever leads in the degradation of energy to its manifestation in
some form of work, at a stage short of that ultimate degradation
which lapses in uniformly diffused heat. This, as Warburg has well
explained, is the general effect or function of the physiological
machine, and in particular of that part of it which we call "cell-

* Precisely as in the Lueretian concursus, motus, ordo, positura, figurae, whereby
bodies mutato ordine mutant naturam.

292 ON THE INTERNAL FORM [ch.

structure*." The normal muscle-cell is something which turns
energy, derived from oxidation, into work; it is a mechanism which
arrests and utihses the chemical energy of oxidation in its downward
course; but the same cell when injured or disintegrated loses its
"usefulness," and sets free a greatly increased proportion of its
energy in the form of heat. It was a saying of Faraday's, that
"even a Hfe is but a chemical act prolonged. If death occur, the
more rapidly oxygen and the affinities run on to their final state f."
Very great and wonderful things are done by means of a
mechanism (whether natural or artificial) of extreme simphcity.
A pool of water, by virtue of its surface, is an admirable mechanism
for the making of waves ; with a lump of ice in it, it becomes an
efficient and self-contained mechanism for the making of currents.
Music itself is made of simple things — a reed, a pipe, a string.
The great cosmic mechanisms are stupendous in their simphcity;
and, in point of fact, every great or little aggregate of heterogeneous
matter (not identical in "phase") involves, ipso facto, the essentials
of a mechanism. Even a non-living colloid, from its intrinsic hetero-
geneity, is in this sense a mechanism, and one in which energy is
manifested in the movement and ceaseless rearrangement of the
constituent particles. For this reason Graham speaks somewhere
or other of the colloid state as "the dynamic state of matter"; in
the same philosopher's' phrase, it possesses ^' energia%.'^

Let us turn then to consider, briefly and diagrammatically, the
structure of the cell, a fertilised germ-cell or ovum for instance, not
in any vain attempt to correlate this structure with the structure
or properties of the resulting and yet distant organism ; but merely
to see how far, by the study of its form and its changing internal
configuration, we may throw hght on certain forces which are for
the time being at work within it.

We may say at once that we can scarcely hope to learn more of
these forces, in the first instance, than a few facts regarding their

* Otto Warburg, Beitrage zur Physiologie der Zelle, insbesondere iiber die
Oxidationsgeschwindigkeit in Zellen; in Asher-Spiro's Ergebnisse der Physiologie,
XIV, pp. 253-337, 1914 (see p. 315).

t See his Life by Bence Jones, ii, p. 299.

X Both phrases occur, side by side, in Graham's classical paper on Liquid
diffusion applied to analysis, Phil. Trans, cli, p. 184, 1861; Chem. and Phys.
Researches (ed. Angus Smith), 1876, p. 554.

iv] AND STRUCTURE OF THE CELL 293

direction and magnitude; tli£ nature and specific identity of the
force or forces is a very different matter. This latter problem is
likely to be difficult of elucidation, for the reason, among others,
that very different forces are often much alike in their outward and
visible manifestations. So it has come to pass that we have a
multitude of discordant hypotheses as to the nature of the forces
acting within the cell, and producing in cell division the "caryo-
kinetic" figures of which we are about to speak. One student may,
like Rhumbler, choose to account for them by an hj^othesis of
mechanical traction, acting on a reticular web of protoplasm*;
another, hke Leduc, may shew us how in many of their most striking
features they may be admirably simulated by salts diffusing in a
colloid medium; others, hke Lamb and Graham Cannon, have
compared them to the stream-hnes produced and the field of force
set up by bodies vibrating in a fluid; others, like Gallardof and
Rhumbler in his earher papers J, insisted on their resemblance to
certain phenomena of electricity and magnetism §; while Hartog
believed that the force in question is only analogous to these, and
has a specific identity of its own||. All these conflicting views are
of secondary importance, so long as we seek only to account for
certain configurations which reveal the direction, rather than the
nature, of a force. One and the same system of lines of force may
appear in a field of magnetic or of electrical energy, of the osmotic
energy of diffusion, of the gravitational energy of a flowing stream.
In short, we may expect to learn something of the pure or abstract
dynamics long before we can deal with the special physics of the

* L. Rhumbler, Mechanische Erklarung der Aehnlichkeit zwischen magne-
tischen Kraftliniensystemen und Zelltheilungsfiguren, Arch. f. Entw. Mech. xv,
p. 482, 1903.

t A. Gallardo, Essai d'interpretation des figures caryocinetiques, Armies del
Museo de Buenos- Aires (2), ii, 1896; Arch. f. Entw. Mech. xxvm, 1909, etc.

X Arch.f. Entw. Mech. ill, iv, 1896-97.

§ On various theories of the mechanism of mitosis, see (e.g.) Wilson, The Cell
in Development, etc.; Meves, Zelltheilung, in Merkel u. Bonnet's Ergehnisse der
Anatomic, etc., vii, viii, 1897-98; Ida H. Hyde, Amer. Journ. Physiol, xii, pp. 241-
275, 1905; and especially A. Prenant, Theories et interpretations physiques de
la mitose, Journ. de VAnat. et Physiol, xlvi, pp. 511-578, 1910. See also A. Conard,
Sur le mecanisme de la division cellulaire, et sur les bases, morphologiques de la
Cytologie, Bruxelles, 1939: a work which I find hard to follow.

II M. Hartog, Une force nouvelle: le mitokinetisme, C.R. 11 Juli 1910; Arch. f.
Entw. Mech. xxvii, pp. 141-145, 1909; cf. ibid, xl, pp. 33-64, 1914.

294 ON THE INTERNAL FORM [ch.

cell. For indeed, just as uniform expansion about a single centre,
to whatsoever physical cause it may be due, will lead to the con-
figuration of a sphere, so will any two centres or foci of potential
(of whatsoever kind) lead to the configurations with which Faraday
first made us famihar under the name of "lines of force*"; a^d
this is as much as to say that the phenomenon, though physical in the
concrete, is in the abstract purely mathematical, and in its very essence
is neither more nor less than a property of three-dimensional space.

But as a matter of fact, in this instance, that is to say in trying
to explain the leading phenomena of the caryokinetic division of
the cell, we shall soon perceive that any explanation which is based,
like Rhumbler's, on mere mechanical traction, is obviously inade-
quate, and we shall find ourselves limited to the hypothesis of some
polarised and polarising force, such as we deal with, for instance,
in magnetism or electricity, or in certain less familiar phenomena
of hydrodynamics. Let us speak first of the cell itself, as it appears
in a state of rest, and let us proceed afterwards to study the more
active phenomena which accompany its division.

Our typical cell is a spherical body; that is to say, the uniform
surface-tension at its boundary is balanced by the outward resistance
of uniform forces within. But at times the surface-tension may be
a fluctuating quantity, as when it produces the rhythmical con-
tractions or "Ransom's waves "f on the surface of a trout's egg; or
again, the surface-tension may be locally unequal and variable, giving
rise to an amoeboid figure, as in the egg of HydraX-

Within the cell is a nucleus or germinal vesicle, also spherical,

* The configurations, as obtained by the usual experimental methods, were
of course known long before Faraday's day, and constituted the "convergent and
divergent magnetic curves" of eighteenth century mathematicians. As Leslie
said, in 1821, they were "regarded with wonder by a certain class of dreaming
philosophers, whp did not hesitate to consider them as the actual traces of an
invisible fluid, perpetually circulating between the poles of the magnet." Faraday's
great advance was to interpret them as indications of stress in a medium — of.
tension or attraction along the lines, and of repulsion transverse to the lines, of the
diagram.

t W. H. Ransom, On the ovum of osseous fishes, Phil. Trans, clvii, pp. 431-502,
1867 (vide p. 463 et. seq.) (Ransom, afterwards a Nottingham physician, was
Huxley's friend and class-fellow at University College, and beat him for the medal
in Grant's class of zoology.)

X Cf, also the curiou.s phenomenon in a dividing egg described as "spinning"
by Mrs G. F. Andrews, Journ. Morph. xii, pp. 367-389, 1897.

IV] AND STRUCTURE OF THE CELL 295

and consisting of portions of "chromatin," aggregated together
within a more fluid drop. The fact has often been commented
upon that, in cells generally, there is no correlation of form
(though there apparently is of size) between the nucleus and the
"cytoplasm," or main body of the cell. So Whitman* remarks
that "except during the process of division the nucleus seldom
departs from it^ cypical spherical form. It divides and sub-divides,
ever returning to the same round or oval form. . . . How different
with the cell. It preserves the spherical form as rarely as the
nucleus departs from it. Variation in form marks the beginning
and the end of every important chapter in its history." On simple
dynamical grounds, the contrast is easily explained. So long as
the fluid substance of the nucleus is qualitatively different from,
and incapable of mixing with, the fluid or semi-fluid protoplasm
surrounding it, we shall expect it to be, as it almost always is, of
spherical form. For on the one hand, it has a surface of its own
whose surface-tension is presumably uniform, and on the other, it
is immersed in a medium which transmits on all sides a uniform
fluid or "hydrostatic" pressure f; thus the case of the spherical
nucleus is closely akin to that of the spherical yolk within the
bird's egg. Again, for a similar reason, the contractile vacuole of
a protozoon is spherical {. It is just a drop of fluid, bounded by a

* Whitman, Journ. Morph. ii, p. 40, 1889.

t "Souvent il n'y a qu'une separation physique entre le cytoplasme et le sue
hucleaire, comme entre deux liquides immiscibles, etc."; Alexeieff, 8ur la mitose
dite primitive, Arch. f. Protistenk. xxix, p. 357, 1913.

X The appearance of " vacuolation " is a result of endosmosis, or the diffusion
of a less dense fluid into the denser plasma of the cell. But while water is probably
taken up at the surface of the cell by purely passive osmotic intake, a definite
"vacuole" appears at a place where osmotic work is being actively done. A higher
osmotic pressure than that of the external medium is maintained within the cell,
but as a "steady state" rather than a condition of equilibrium, in other words by
the continual expenditure of energy; and the difference of pressure is at best small.
The "contractile vacuole" bursts when it touches the surface of the cell, and
bursting may be delayed by manipulating the vacuole towards the interior. It
may sometimes burst towards the interior of the cell through inequalities in its
own surface-tension, and the collapsing vacuole is then apt to shew a star-shaped
figure. The cause of the higher osmotic pressure within the cell is a matter for
the colloid chemist, and cannot be discussed here. On the physiology of the
contractile vacuole, see {int. al.) H. Z. Gow, Arch. f. Protistenk. lxxxvii, pp. 185-
212, 1936; J. Spek, Einfluss der Salze auf die Plasmkolloide von Actinosphaerium,
Acta Zool. 1921; J. A, Kitching, Journ. Exp. Biology, xi, xiii, xv, 1934-38.

296 ON THE INTERNAL FORM [ch.

uniform surface-tension, and through whose boundary-film diffusion
is taking place; but here, owing to the small difference between the
fluid constituting and that surrounding the drop, the surface-tension
equihbrium is somewhat unstable; it is apt to vanish, and the
rounded outhne of the drop disappears, Uke a burst bubble, in a
moment.

If, on the other hand, the substance of the cell acquire a greater
soUdity, as for instance in a muscle-cell, or by reason of mucous
accumulations in an epithehum cell, then the laws of fluid pressure
no longer apply, the pressure on the nucleus tends to become
unsymmetrical, and its shape is modified accordingly. Amoeboid
movements may be set up in the nucleus by anything which disturbs
the symmetry of its own surface-tension; and where "nuclear
material" is scattered in small portions throughout the cell as in
many Rhizopods, instead of being aggregated in a single nucleus,
the simple explanation probably is that the "phase difference" (as
the chemists say) between the nuclear and the protoplasmic substance
is comparatively shght, and the surface-tension which tends to keep
them separate is correspondingly small*.

Apart from that invisible or ultra-microscopic heterogeneity
which is inseparable from our notion of a "colloid," there is a
visible heterogeneity of structure within both the nucleus and the
outer protoplasm. The former contains, for instance, a rounded
nucleolus or "germinal spot," certain conspicuous granules or
strands of the peculiar substance called chromatin*)*, and a coarse
mesh work of a protoplasmic material known as "linin" or achro-
matin; the outer protoplasm, or cytoplasm, is generally believed
to consist throughout of a sponge-work, or rather alveolar mesh-
work, of more and less fluid substances; it may contain "mito-
chondria," appearing in tissue-cultures as small amoeboid bodies;
and lastly, there are generally to be detected (in the animal, rarely
in the vegetable kingdom) one or more very minute bodies, usually
in the cytoplasm sometimes within the nucleus, known as the
centrosome or centrosomes.

* The elongated or curved "macronucleus" of an Infusorian is to be looked
upon as a single mass of chromatin, rather than as an aggregation of particles in
a fluid drop, as in the case described. It has a shape of its own, in which ordinary
surface-tension plays a very subordinate part.

•j- First so-called by W. Flemming, in his Zellsubstanz, Kern und Zelltheilung, 1882.

IV] AND STRUCTURE OF THE CELL 297

The morphologist is accustomed to speak of a "polarity" of the
cell, meaning thereby a symmetry of visible structure about a
particular axis. For instance, whenever we can recognise in a cell
both a nucleus and a centrosome, we may consider a hne drawn
through the two as the morphological axis of polarity ; an epitheUum
cell is morphologically symmetrical about a median axis passing
from its free surface to its attached base. Again, by an extension
of the term polarity, as is customary in dynamics, we may have
a "radial" polarity, between centre and periphery; and lastly, we
may have several apparently independent centres of polarity within
the single cell. Only in cells of quite irregular or amoeboid form
do we fail to recognise a definite and symmetrical polarity. The
morphological polarity is accompanied by, and is but the outward
expression (or part of it) of a true dynamical polarity, or distribution
of forces; and the hues of force are, or may be, rendered visible
by concatenation of particles of matter, such as come under the
influence of the forces in action.

When hnes of force stream inwards from the periphery towards
a point in the interior of the cell, particles susceptible of attraction
either crowd towards the surface of the cell or, when retarded by
friction, are seen forming lines or "fibrillae" which radiate outwards
from the centre. In the cells of columnar or cihated epithehum,
where the sides of the cell are symmetrically disposed to their
neighbours but the free and attached surfaces are very diverse from
one another in their external relations, it is these latter surfaces
which constitute the opposite poles; and in accordance with the
parallel lines of force so set up, we very frequently see parallel lines
of granules which have ranged themselves perpendicularly to the
free surface of the cell (cf. Fig. 149).

A simple manifestation of polarity may be well illustrated by
the phenomenon of diffusion, where we may conceive, and may
automatically reproduce, a field of force, with its poles and its
visible lines of equipotential, very much as in Faraday's conception
of the field of force of a magnetic system. Thus, in one of Leduc's
experiments*, if we spread a layer of salt solution over a level
plate of glass, and let fall into the middle of it a drop of indian
ink, or of blood, we shall find the coloured particles travelling
* Thtorie physico-chimique de la Vie, 1910, p. 73.

298 ON THE INTERNAL FORM [ch

outwards from the central "pole of concentration" along the lines
of diffusive force, and so mapping out for us a "monopolar field"
of diffusion : and if we set two such drops side by side, their fines
of diffusion will oppose and repel one another. Or, instead of the
uniform layer of salt solution, we may place at a little distance
from one another a grain of salt and a drop of blood, representing
two opposite poles: and so obtain a picture of a "bipolar field"
of diffusion. In either case, we obtain results closely analogous to
the morphological, but really dynamical, polarity of the organic
cell. But in all probability, the dynamical polarity or asymmetry
of the cell is a very complicated phenomenon: for the obvious
reason that, in any system, one asymmetry will tend to beget
another. A chemical asymmetry will induce an inequafity of
surface-tension, which will lead directly to a modification of form ;
the chemical asymmetry may in turn be due to a process of
electrolysis in a polarised electrical field; and again the chemical
heterogeneity may be intensified into a chemical polarity, by the
tendency of certain substances to seek a locus of gi^eater or less
surface-energy. We need not attempt to grapple with a subject so
compHcated, and leading to so many problems which lie beyond
the sphere of interest of the morphologist. But yet the morpho-
logist, in his study of the cell, cannot quite evade these important
issues; and we shall return to them again when we have dealt
somewhat with the form of the cell, and have taken account of
some of its simpler phenomena.

We are now ready, and in some measure prepared, to study the
numerous and complex phenomena which accompany the division
of the cell, for instance of the fertilised egg. But it is no easy task
to epitomise the facts of the case, and none the easier that of late
new methods have shewn us new things, and have cast doubt on
not a little that we have been accustomed to believe.

Division of the cell is of necessity accompanied, or preceded, by
a change from a radial or monopolar to a definitely bipolar sym-
metry. In the hitherto quiescent or apparently quiescent cell, we
perceive certain movements, which correspond precisely to what
must accompany and result from a polarisation of forces within:
of forces which, whatever be their specific nature, are at least

IV] AND STRUCTURE OF THE CELL 299

capable of polarisation, and of producing consequent attraction or
repulsion between charged particles. The opposing forces which
are distributed in equilibrium throughout the cell become focused
in two "centrosomes*," which may or may not be already visible.
It generally happens that, in the egg, one of these centrosomes is
near to and the other far from the "animal pole," which is both
visibly and chemically different from the other, and is where the
more conspicuous developmental changes will presently begin.

Between the two centrosomes, in stained preparations, a spindle-
shaped figure appears (Fig. 88), whose striking resemblance to the

Fig. 8». Caryokinetic ligure in a dividing cell (or blastomere) of a trout s egg.
After Prenant, from a preparation by Prof. Bouin.

lines of force made visible by iron-filings between the poles of a
magnet was at once recognised by Hermann Fol, in 1873, when he
witnessed the phenomenon for the first timef. On the farther
side of the centrosomes are seen star-like figures, or "asters," in
which we se^m to recognise the broken lines of force which run
externally to those stronger lines which lie nearer to the axis and
constitute the "spindle." The lines of force are rendered visible,
or materialised, just as in the experiment of the iron-fihngs, by the
fact that, in the heterogeneous substance of the cell, certain portions

* These centrosomes are the two halves of a single granule, and are said (by
Boveri) to come from the middle piece of the original spermatozoon.

t He did so in the egg of a medusa {Geryon), Jen. Zeitschr. vii, p. 476, 1873.
Similar ideas have been expressed by Strasbiirger, Henneguy, Van Beneden,
Errera, Ziegler, Gallardo and others.

300 ON THE INTERNAL FORM [ch.

of matter are more "permeable" to the acting force than others,
become themselves polarised after the fashion of a magnetic or
"paramagnetic" body, arrange themselves in an orderly way
between the two poles of the field of force, seem to cling to one
another as it were in threads*, and are only prevented by the
friction of the surrounding medium from approaching and con-
gregating around the adjacent poles.

As the field of force strengthens, the more will the lines of force
be drawn in towards the interpolar axis, and the less evident will
be those remoter lines which constitute the terminal, or extrapolar,
asters: a clear space, free from materialised fines of force, may
thus tend to be set up on either side of the spindle, the so-called
''Biitschfi space" of the histologistsl. On the other hand, the lines
of force constituting the spindle will be less concentrated if they
find a path of less resistance at the periphery of the ceU : as happens
in our experiment of the iron-filings, when we encircle the field of
force with an iron ring. On this principle, the differences observed
between cells in which the spindle is well developed and the asters
small, and others in which the spindle is weak and the asters greatly
developed, might easily be explained by variations in the potential
of the field, the large, conspicuous asters being correlated in turn
with a marked permeability of the surface of the cell.

The visible field of force, though often called the "nuclear
spindle," is formed outside of, but usually near to, the nucleus.

* Whence the name "mitosis" (Greek /ziros, a thread), applied first by Flemming
to the whole phenomenon. Kolimann (Biol. Centralbl. ii, p. 107, 1882) called it
divisio per fila, or divisio laqueis implicata. Many of the earlier students, such as
Van Beneden (Rech. sur la maturation de I'oeuf, Arch, de Biol, iv, 1883), and
Hermann Fol (Zur Lehre v. d. Entstehung d. karyokinetischen Spindel, Arch. f.
mikrosk. Anat. x,xxvii, 1891) thought they recognised actual muscular threads,
drawing the nuclear material asunder towards the respective foci or poles; and
some such view of Zugkrdfte was long maintained by other writers, by Heidenhain
especially, by Boveri, Flemming, R. Hertwig, Rhumbler, and many more. In fact,
the existence of contractile threads, or the ascription to the spindle rather than to
the poles or centrosomes of the active forces concerned in nuclear division, formed
the main tenet of all those who declined to go beyond the "contractile properties
of protoplasm" for an explanation of the phenomenon (cf. J. W. Jenkinson,
Q.J. M.S. XLViii, p. 471, 1904. See also J. Spek's historical account of the theories
of cell-division. Arch. f. Entw. Mech. xliv, pp. 5-29, 1918).

t Cf. 0, Biitschli, Ueber die kiinstliche Nachahmung der karyokinetischen
Figur, Verh. Med. Nat. Ver. Heidelberg, v, pp. 28-41 (1892), 1897.

IV

AND STRUCTURE OF THE CELL

301

Let us look a little more closely into the structure of this body,
and into the changes which it presently undergoes.

Within its spherical outHne (Fig. 89 a), it contains an ''alveolar"
meshwork (often described, from its appearance in optical section,
as a "reticulum"), consisting of more sohd substances with more
fluid matter filling up the interalveolar spaces. This phenomenon,
familiar to the colloid chemist, is what he calls a "two-phase
system," one substance or "phase" forming a continuum through
which the other is dispersed; it is closely alhed to what we call in

atr racHon - sphere
1 ,C€ntro3omes

Fig. 89 A.

Fig. 89 B.

ordinary language a. froth ot'sl foam*, save that in these latter the
disperse phase is represented by air. It is a surface-tension pheno-
menon, due to the interaction of two intermixed fluids not very
different in density, as they strive to separate. Of precisely the
same kind (as Biitschli was the first to shew) are the minute alveolar
networks which are to be discerned in the cytoplasm of the cellf,

* Froth and foam have been much studied of late years for technical reasons,
and other factors than surface-tension are foiind to be concerned in their existence
and their stability. See (int. al.) Freundlich's Capillarchemie, and various papers
by Sasaki, in Bull. Chem. Soc. of Japan, 1936-39.

t Biitschli, Untersuchungen iiber mikroskopische Schdume und das Protoplasma,
1892; Untersuchungen uber Strukturen, etc., 1898; L. Rhumbler, Protoplasma als
physikalisches System, Ergehn. d. Physiologie, 1914; H. Giersberg, Plasmabau
der Amoben, im Hinblick auf die Wabentheorie, Arch. f. Entw. Mech. li, pp. 150-250,
1922; etc.

302 ON THE INTERNAL FORM [ch.

and which we now know to be not inherent in the nature of proto-
plasm nor of Hving matter in general, but to be due to various
causes, natural as well as artificial*. The microscopic honeycomb
structure of cast metal under various conditions of coohng is an
example of similar surface-tension phenomena.

Such then, in briefest outhne, is the typical structure commonly
ascribed to a cell when its latent energies are about to manifest
themselves in the phenomenon of cell-division. The account is
based on observation not of the hving cell but of the dead : on the
assumption, that is to say, that fixed and stained material gives a
true picture of reahty. But in Robert Chambers's method of micro-
dissection f, the hving cell is manipulated with fine glass needles
under a high magnification, and shews us many interesting things.
Chambers assures us that the spindle fibres never make their
appearance as visible structures until coagulation has set in; and
that astral rays are, or appear to be, channels in which the more
fluid content of the cell flows towards a centrosome:|:. Within the
bounds to which we are at present keeping, these things are of no
great moment; for whether the spindle appear early or late, it still
bears witness to the fact that matter has arranged itself along
bipolar fines of force; and even if the astral rays be only streams
or currents, on lines of force they still approximately he. Yet the
change from the old story to the new is important, and may make
a world of diff'erence when we attempt to define the forces concerned.
All our descriptions, all our interpretations, are bound to be
influenced by our conception of the mechanism before us; and he

* Arrhenius, in describing a typical colloid precipitate, does so in terms that
are very closely applicable to the ordinary microscopic appearance of the protoplasm
of the cell. The precipitate consists, he says, "en un reseau d'une substance solide
contenant peu d'eau, dans les mailles duquel est inclus un fluide contenant un peu
de colloide dans beaucoup d'eau. . . . Evidemment cette structure se forme a cause
de la petite difference de poids specifique des deux phases, et de la consistance
gluante des particules separees, qui s'attachent en forme de reseau " {Rev. Scientifique,
Feb. 1911). This, however, is far from being the whole story: cf. (e.g.) S. C.
Bradford, On the theory of gels, Biochem. Journ. xvii, p. 230, 1925; W. Seifritz,
The alveolar structure of protoplasm, Protoplasma, ix, p. 198, 1930; and A. Frey-
Wissling, Submikroskopische Morphologie des Protoplasmas, Berlin, 1938.

t See R. Chambers, An apparatus. . .for the dissection and injection of living
cells, Anatom. Record, xxiv, 19 pp., 1922.

X This centripetal flow of fluid was announced by Biitschli in his early papers,
and confirmed by Rhumbler, though attributed to another cause.

IV] AND STRUCTURE OF THE CELL 303

who sees threads where another sees channels is hkely to tell a
different story about neighbouring and associated things.

It has also been suggested that the spindle is somehow due to a re-arrange-
ment of protein macromolecules or micelles ; that such changes of orientation
of large colloid particles may be a widespread phenomenon; and that coagu-
lation itself is but a polymerisation of larger and larger macromolecules*.

But here we have touched the brink of a subject so important that we must
not pass it by without a word, and yet so contentious that we must not enter
into its details. The question involved is simply whether the great mass of
recorded observations and accepted beliefs with regard to the visible structure
of protoplasm and of the cell constitute a fair picture of the actual living cell,
or be based on appearances which are incident to death itself and to the
artificial treatment which the microscopist is accustomed to apply. The great
bulk of histological work is done by methods which involve the sudden killing
of the cell or organism by strong reagents, the assumption being that death
is so rapid that the visible phenomena exhibited during life are retained or
"fixed" in our preparations.

Hermann Fol struck a warning note full sixty years ago: "II importe a
I'avenir de I'histologie de combattre la tendance a tirer des conclusions des
images obtenues par des moyens artificiels et a leur donner une valeur intrin-
seque, sans que ces images aient ete controlees sur le vivantf." Fol was
thinking especially of cell-membranes and the delimitation of cells; but still
more difficult and precarious is the interpretation of the minute internal net-
works, granules, etc., which represent the alleged structure of protoplasm.
A colloid body, or colloid solution, is ipso facto heterogeneous; it has after
some fashion a structure of its own. And this structure chemical action,
under the microscope, may demonstrate, or emphasise, or alter and disguise.
As Hardy put it, "It is notorious that the various fixing reagents are co-
agulants of organic colloids, and that the figure varies according to the reagent
used."

A case in point is that of the vitreous humour, to which some histologists
have ascribed a fairly complex structure, seeing in it a framework of fibres
with the meshes filled with fluid. But it is really a true gel, without any
structure in the usual sense of the word. The "fibres" seen in ordinary
microscopic preparations are due to the coagulation of micellae by the fixative
employed. Under the ultra -microscope the vitreous is optically empty to
begin with; then innumerable minute fibrillae appear in the beam of light,
criss-crossing one another. Soon these break down into strings of beads, and

* Cf. J. D. Bernal, on Molecular architecture of biological systems, Proc. Boy.
Inst., 1938; H. Staiidinger, Nature, Aug. I, 1939.

t H. Fol, Becherches sur la fecondation et le commencement de VMnogenie chez
divers animaux, Geneve, 1879, pp. 241-242. Cf. A. Daleq, in Biol. Beviews, iii,
p. 24, 1928: "II serait desirable de nous debarrasser de I'idee que tout ce qu'il
y a d'important dans la cellule serait providentiellement colorable par I'hematoxy-
line, la safranine ou le violet de gentiane."

304 ON THE INTERNAL FORM [ch.

finally only separate dots are seen*. Other sources of error arise from the
optical principles concerned in microscopic vision; for the diffraction-pattern
which we call the "image" may, under certain circumstances, be very different
from the actual object f. Furthermore, the optical properties of living proto-
plasm are especially complicated and imperfectly known, as in general those
of colloids may be said to be; the minute aggregates of the "disperse phase"
of gels produce a scattering action on light,' leading to appearances of turbidity
etc., with no other or more real basis J.

So it comes to pass that some writers have altogether denied the existence
in the living cell-protoplasm of a network or alveolar "foam"; others have
cast doubts on the main tenets of recent histology regarding nuclear structure ;
and Hardy, discussing the structure of certain gland-cells, declared that
"there is no evidence that the structure discoverable in the cell-substance of
these cells after fixation has any counterpart in the cell when living." "A
large part of it" he went on to say "is an artefact. The profound difference
in the minute structure of a secretory cell of a mucous gland according to the
reagent which is used to fix it would, it seems to me, almost suffice to establish
this statement in the absence of other evidence §."

Nevertheless, histological study proceeds, especially on the part of the
morphologists, with but little change in theory or in method, in spite of these
and many other warnings. That certain visible structures, nucleus, vacuoles,
"attraction-spheres" or centrosomes, etc., are actually present in the living
cell we know for certain; and to this class belong the majority of structures
with which we are at present concerned. That many other alleged structures
are artificial has also been placed beyond a doubt; but where to draw the
dividing line we often do not know.

The following is a brief epitome of the visible changes undergone
by a t3rpical cell, subsequent to the resting stage, leading up to the
act of segmentation, and constituting the phenomenon of mitosis
or caryokinetic division. In the fertilised egg of a sea-urchin we
see with almost diagrammatic completeness, in fixed and stained
specimens, what is set forth here||.

* W. S. Duke-Elder, Journ. Physiol, lxviii, pp. 1.54-165, 1930; of. Baurmann,
Arch.f. Ophthalm. 1923, 1926; etc.

t Abbe, Arch.f. mikrosk. Anat. ix, p. 413, 1874; Gesammelte Ahhandl. i, p. 45,
1904.

X Cf. Rayleigh, On the light from the sky, Phil. Mag. (4) xli, p. 107, 1871.

§ W. B. Hardy, On the structure of cell protoplasm, Journ. Physiol, xxiv,
pp. 158-207, 1889; also Hober, Physikalische Chemie der Zelle und der Gewebe,
1902; W. Berg, Beitrage zur Theorie der Fixation, etc., Arch.f. mikr. Anat. LXii,'
pp. 367-440, 1903, Cf. {int. al.) Flemming, Zellsubstanz, Kern und Zelltheilung,
1882, p. 51; etc.

II My description and diagrams (Figs. 89-93) are mostly based on those of
the late Professor E. B. Wilson.

IV]

AND STRUCTURE OF THE CELL

305

1. The chromatin, which to begin with had been dimly seen as
granules on a vague achromatic reticulum (Figs. 89, 90) — perhaps no
more than an histological artefact — concentrates to form a skein or
spireme, often looked on as a continuous thread, but perhaps
discontinuous or fragmented from the first. It, or its several
fragments, will presently spht asunder; for it is essentially double,
and may even be seen as a double thread, or pair of chromatids, from
an early stage. The chromosomss are portions of this double thread,
which shorten down to form httle rods,' straight or curved, often

chromoaome*

Fig. 90 A.

Fig. 90 B.

bent into a V, sometimes ovoid, round or even annular, and which
in the living cell are frequently seen in active, writhing movement,
*"hke eels in a box"*; they keep apart from one another, as by
some repulsion, and tend to move outward towards the nuclear
membrane. Certain deeply staining masses, the nucleoh, may be
present in the resting nucleus, but take no part (at least as a rule)
in the formation of the chromosomes; they are either cast out of
the nucleus and dissolved in the cytoplasm, or else fade away in situ.

* T. S. Strangeways, Proc. E.S. (B), xciv, p. 139, 1922. The tendency of the
chromatin to form spirals, large or small, while the nucleus is issuing from its
resting-stage, is very remarkable. The tensions to which it is due may be overcome,
and the chromosomes made to uncoil, by treatment with ammonia or acetic acid
vapour. See Y. Kuwada, Botan. Mag. Tokyo, xlvi, p. 307, 1932; and C. D.
Darlington, Mechanical aspects of nuclear division, Sci. Journ. B. Coll, of Sci.
TV, p. 94, 1934.

306 ON THE INTERNAL FORM [ch.

But this rule does not always hold; for they persist in many
protozoa, and now and then the nucleolus remains and becomes
itself a chromosome, as in the spermogonia of certain insects.

2. Meanwhile a certain deeply staining granule (here extra-
nuclear), known as the centrosome*, has divided into two. It is all
but universally visible, save in the higher plants; perhaps less stress
is laid on it than at one time, but Bovery called i-t the "dynamic
centre" of the cellf. The two resulting granules travel to
opposite poles Df the nucleus, and there eacii becomes surrounded
by a starhke figure, the aster, of which we have sptken already;
immediately around the centrosome is a clear space, the centro-
sphere. Between the two centrosomes, or the two asters, stretches
the spindle. It lies in the long axis, if there be one, of the cell, a
rule laid down nearly sixty years ago, and still remembered as
"Hertwig's Law" J; but the rule is as much and no more than to
say that the spindle sets in the direction of least resistance. Where
the egg is laden with food-yolk, as often happens, the latter is
heavier than the cytoplasm; and gravity, by orienting the egg
itself, thus influences, though only indirectly, the first planes of
segmentation §.

3. The definite nuclear outhne is soon lost; for the chemical
"phase-difference" between nucleus and cytoplasm has broken
down, and where the nucleus was, the chromosomes now he (Figs.
90, 91). The lines of the spindle become visible, the chromosomes
arrange themselves midway between its poles, to form the equatorial
plate, and are spaced out evenly around the central spindle, again
a simple result of mutual repulsion.

4. Each chromosome separates longitudinally into two|| : usually
at this stage — but it is to be noted that the spHtting may have taken
place as early as the spireme stage (Fig. 92).

* The centrosome has a curious history of its own, none too well ascertained.
The ovum has a centrosome, and in self- fertilised eggs this is retained; but when
a sperm-cell enters the egg the original centrosome degenerates, and its place is
taken by the "middle-piece" of the spermatozoon,

f The stages 1, 2, 5 and 6 are called by embryologista the prophase, metaphase,
anaphase and telophase.

X C. Hertwig, Jenaische Ztschr. xviii, 1884.

§ See James Gray, The effect of gravity on the eggs of Echinus, Jl. Exp. Zool. v,
pp. 102-11, 1927.

II A fundamental fact, first seen by Flemming in 1880.

IV]

AND STRUCTURE OF THE CELL

307

5. The halves of the spht chromosomes now separate from and
apparently repel one another, travelUng in opposite directions
towards the two poles* (Fig. 92 b), for all the world as though they
were being pulled asunder by actual threads.

Fig. 91 A.

Fig. 91 B.

central spindle

mantle 'fibres

split chromosome*

Fig. 92 A.

Fig. 92 B.

6. Presently the spindle itself changes shape, lengthens and con-
tracts, and seems as it were to push the two groups of daughter-

* Cf. K. Belar, Beitrage zur Causalanalyse der Mitose, Ztschr. f. Zellforschung,
X, pp. 73-124, 1929.

308

ON THE INTERNAL FORM

[CH.

chromosomes into their new places* (Figs. 92, 93); and its chromo-
somes form once more an alveolar reticulum and may occasionally
form another spireme at this stage. A boundary-surface, or at least a
recognisable phase-difference, now develops round each reconstructed
nuclear mass, and the spindle disappears (Fig. 93 b). The centrosome
remains, as a rule, outside the nucleus.

7. On the central spindle, in the position of the equatorial plate,
a "cell-plate," consisting of deeply staining thickenings, has made
its appearance during the migration of the chromosomes. This cell-
plate is more conspicuous in plant-cells.

otfrottron spncft

,d>iopptQr,ng spmifte

Htcon^rucffd dough/trnuei^t

Fig. 93 B.

8. Meanwhile a constriction has appeared in the cytoplasm, and
the cell divides through the equatorial plane. In plant-cells the
line of this division is foreshadowed by the "cell-plate," which
extends from the spindle across the entire cell, and spHts into two
layers, between which appears the membrane by which the daughter-
cells are cleft asunder. In animal cells the cell-plate does not attain
such dimensions, and no cell-wall is formed.

The whole process takes from half-an-hour to an hour; and this
extreme slowness is not. the least remarkable part of the pheno-
menon, from a physical point of view. The two halves of the

* The spindle has no actual threads or fibres, for Robert Chambers's micro-
needles pass freely through it without disturbing the chromosomes: nor is it
visible at all in living cells in vitro. It seems to be due to partial gelation of the
cytoplasm, under conditions which, whether they be mechanical or chemical, are
not easy to understand.

IV] AND STRUCTURE OF THE CELL 309

dividing centrosome, while moving apart, take some twenty minutes
to travel a distance of 20 /x, or at the rate, say, of two years to a
yard. It is a question of inertia, and the inertia of the system must
be very large.

The beautiful technique of cell-culture in vitro has of late years
let this whole succession of phenomena, once only to be deduced
from sections, be easily followed as it proceeds within the living
tissue or cell. The vivid accounts which have been given of this
spectacle add little to the older account as we have related it:
save that, when the equatorial constriction begins and the halves
of the split chromosomes drift apart, the protoplasm begins to show
a curious and even violent activity. The cytoplasm is thrust in
and out in bulging pustules or "balloons"; and the granules and
fat-globules stream in and out as the pustules rise and fall away.
At length the turmoil dies down; and now each half of the cell
(not an ovum but a tissue-cell or "fibroplast") pushes out large
pseudopodia, flattens into an amoeboid phase, the connecting thread
of protoplasm snaps in the divided cell, and the daughter-cells fall
apart and crawl away. The two groups of chromosomes, on reaching
the poles of the spindle, turn into bunches of short thick rods; these
grow diffuse, and form a network of chromatin within a nucleus;
and at last the chromosomes, having lost their identity, disappear
entirely, and two or more nucleoH are all that is to be seen within
the cell.

The whole, or very nearly the whole, of these nuclear phenomena
may be brought into relation with some such polarisation of forces
in the cell as a whole as is indicated by the "spindle" and "asters"
of which we have already spoken: certain particular phenomena,
directly attributable to surface-tension and diffusion, taking place
in more or less obvious and inevitable dependence upon the polar
system. At the same time, in attempting to explain the phenomena,
we cannot say too clearly, or too often, that all that we are meanwhile
justified in doing is to try to shew that such and such actions He
within the range of known physical actions and phenomena, or that
known physical phenomena produce effects similar to them. We
feel that the whole phenomenon is iiot sui generis, but is some-
how or other capable of being referred to dynamical laws, and to

310 ON THE INTERNAL FORM [ch.

the general principles of physical science. But when we speak of
some particular force or mode of action, using it as an illustrative
hypothesis, we stop far short of the implication that this or that
force is necessarily the very one which is actually at work within
the living cell; and certainly we need not attempt the formidable
task of trying to reconcile, or to choose between, the various
hypotheses which have already been enunciated, or the several
assumptions on which they depend.

Many other things happen within the cell, especiall)^ in the germ-
cell both before and after fertilisation. They also have a physical
element, or a mechanical aspect, like the phenomena of cell-
division which we are speaking of; but the narrow bounds to which
we are keeping hold difficulties enough*.

Any region of space within which action is manifested is a field
of force; and a simple example is a bipolar field, in which the
action is symmetrical with reference to the fine joining two points,
or poles, and with reference also to the "equatorial" plane equi-
distant from both. We have such a field of force in the neigh-
bourhood of the centrosome of the ripe cell or ovum, when it is
about to divide; and by the time the centrosome has divided, the
field is definitely a bipolar one.

The quality of a medium filling the field of force may be uniform,
or it may vary from point to point. In particular, it may depend
upon the magnitude of the field; and the quality of one medium
may differ from that of another. Such variation of quality, within
one medium, or from one medium to another, is capable of diagram-
matic representation by a variation of the direction or the strength
of the field (other conditions being the same) from the state
manifested in some uniform medium taken as a standard. The
medium is said to be permeable to the force, in greater or less degree
than the standard medium, according as the variation of the density
of the lines of force from the standard case, under otherwise identical
conditions, is in excess or defect. A body placed in the medium will
tend to move towards regions of greater or less force according as its

* Cf. C. D. Darlington, JieretU Advances in Cytology, 1932, and other well-known
works.

IV] AND STRUCTURE OF THE CELL 311

penneability is greater or less than that of the surrounding medium'^.
In the common experiment of placing iron-filings between the two
poles of a magnetic field, the filings have a very high permeability;
and not only do they themselves become polarised so as to attract
one another, but they tend to be attracted from the weaker to the
stronger parts of the field, and as we have seen, they would soon
gather together around the nearest pole were it not for friction
or some other resistance. But if we "repeat the same experiment
with such a metal as bismuth, which is very little permeable to the
magnetic force, then the conditions are reversed, and the particles,
being repelled from the stronger to the weaker parts of the field,
tend to take up their position as far from the poles as possible.
The particles have become polarised, but in a sense opposite to that
of the surrounding, or adjacent, field.

Now, in the field of force whose opposite poles are marked by
the centrosomes, we may imagine the nucleus to act as a more or
less permeable body, as a body more permeable than the surrounding
medium, that is to say the " cytoplasm " of the cell. It is accordingly
attracted by, and drawn into, the field of force, and tries, as it
were, to set itself between the poles and as far as possible from both
of them. In other words,' the centrosome-foci will be apparently
drawn over its surface, until the nucleus as a whole is involved
within the field of force which is visibly marked out by the "spindle"
(Fig. 90 b).

If the field of force be electrical, or act in a fashion analogous
to an electrical field, the charged nucleus will have its surface-
tensions diminished f: with the double result that the inner alveolar
mesh work will be broken up (par. 1), and that the spherical
boundary of tKe whole nucleus will disappear (par. 2). The break-
up of the alveoli (by thinning and rupture of their partition walls)

* If the word penneability be deemed too directly suggestive of the phenomena
of magnetism, we may replace it by the more general term of specific iyidiictive
capacity. This would cover the particular case, which is by no means an improbable
one, of our phenomena being due to a "surface charge" borne by the nucleus
itself and also by the chromosomes: this surface charge being in turn the result
of a difference in inductive capacity between the body or particle and its surrounding
medium.

t On the effect of electrical influences in altering the surface-tensions of the
colloid particles, see Bredig, Anorganische Fermente, pp. 15, 16, 1901.

312 ON THE INTERNAL FORM [ch.

leads to the formation of a net, and the further break-up of the net
may lead to the unravelling of a thread or "spireme".

Here there comes into play a fundamental principle which, in
so far as we require to understand it, can be explained in simple
words. The eifect (and we might even say the object) of drawing
the more permeable body in between the poles is to obtain an
"easier path" by which the Hues of force may travel; but it is
obvious that a longer route through the more permeable body may
at length be found less advantageous than a shorter route through
the less permeable medium. That is to say, the more permeable
body will only tend to be drawn into the field of force until a point
is reached where (so to speak) the way round and the way through
are equally advantageous. We should accordingly expect that (on
our hjrpothesis) there would be found cases in which the nucleus
was wholly, and others in which it was only partially, and in greater
or less degree, drawn in to the field between the centrosomes. This
is precisely what is found to occur in actual fact. Figs. 90 a and b
represent two so-called "types," of a phase which follows that
represented in Fig. 89. According to the usual descriptions we are
told that, in such a case as Fig. 90b, the "primary spindle"
disappears* and the centrosomes diverge to opposite poles of the
nucleus; such a condition being found in many plant-cells, and in
the cleavage-stages of many eggs. In Fig. 90 a, on the other hand,
the primary spindle persists, and subsequently comes to form the
main or "central" spindle; while at the same time we see the
fading away of the nuclear membrane, the breaking up of the
spireme into separate chromosomes, and an ingrowth into the nu-
clear area of the "astral rays" — all as in Fig. 91 a, which represents
the next succeeding phase of Fig. 90 b. This condition, of Fig. 91 a,
occurs in a variety of cases; it is well seen in the epidermal cells
of the salamander, and is also on the whole characteristic of the
mode of formation of the "polar bodies t." It is clear and obv;ous
that the two "types" correspond to mere differences of degree,

* The spindle is potentially there, even though (as Chambers assures us) it only
becomes visible after post-mortem coagulation. It is also said to become visible
under crossed nicols: W. J. Schmidt, Biodynamica, xxii, 1936.

t These v/ere first observed in the egg of a pond-snail (Limnaea) by B. Dumortier,
Mim. sur Vemhryoginie des mollusques, Bruxelles, 1837.

IV] AND STRUCTURE OF THE CELL 313

and are such as would naturally be brought about by differences
in the relative permeabilities of the nuclear mass and of the
surrounding cytoplasm, or even by differences in the magnitude of
the former body.

But now an important change takes place, or rather an important
difference appears; for, whereas the nucleus as a whole tended to
be drawn in to the stronger parts of the field, when it comes to break
up we find, on the contrary, that its contained spireme-thread or
separate chromosomes tend to be repelled to the weaker parts.
Whatever this difference may be due to — whether, for instance, to
actual differences of permeability, or possibly to differences in
"surface-charge" or to other causes — the fact is that the chromatin
substance now behaves after the fashion of a "diamagnetic'' body,
and is repelled from the stronger to the weaker parts of the field.
In other words, its particles, lying in the inter-polar field, tend to
travel towards the equatorial plane thereof (Figs. 91, 92), and
further tend to move outwards towards the periphery of that plane,
towards what the histologist calls the "mantle-fibres," or outermost
of the lines of force of which the spindle is made up (par. 5, Fig. 91 b).
And if this comparatively non-permeable chromatin substance come
to consist of separate portions, more or less elongated in form,
these portions, or separate "chromosomes," will adjust themselves
longitudinally, in a peripheral equatorial circle (Figs. 92 a, b). This
is precisely what actually takes place. Moreover, before the breaking
up of the nucleus, long before the chromatin material has broken
up into separate chromosomes, and at the v^ry time when it is
being fashioned into a "spireme," this body already lies in a polar
field, and must already have a tendency to set itself in the equatorial
plane thereof. But the long, continuous spireme thread is unable,
so long as the nucleus retains its spherical boundary wall, to adjust
itself in a simple equatorial annulus; in striving to do so, it must
tend to coil and "kink" itself, and in so doing (if all this be so),
it must t^nd to assume the characteristic convolutions of the
"spireme."

After the spireme has broken up into separate chromosomes,
these bodies come to rest in the equatorial plane, somewhere near
its periphery ; and here they tend to set themselves in a symmetrical
arrangement (Fig. 94), such as makes for still better equihbrium.

314

ON THE INTERNAL FORM

[CH.

The particles ir^y be rounded or linear, straight or bent, sometimes
annular; they may be all alike, or one or more may differ from
the rest. Lying as they do in a semi-fluid medium, and subject
(doubtless) to some symmetrical play of forces, it is not to be
wondered at that they arrange themselves in a symmetrical con-
figuration; and the field of force seems simple enough to let us
predict, to some extent, the symmetries open to them. We do not
know, we cannot safely surmise, the nature of the forces involved.
In discussing Brauer's observations on the sphtting of the chromatic
filament, and on the symmetrical arrangement of the separate
granules, in Ascaris megalocephala, LiUie* remarks: "This behaviour

Fig. 94. Chromosomes, undergoing splitting and separation.
After Hatsehek and Flemming, diagrammatised.

is strongly suggestive of the division of a colloidal particle under
the influence of its surface electrical charge, and of the effects of
mutual repulsion in keeping the products of division apart." It is
probable that surface-tensions between the particles and the sur-
rounding protoplasm would bring about an identical result, and
would sufficiently account for the obvious, and at first sight very
curious symmetry. If we float a couple of matches in water, we
know that they tend to approach one another till they He close
together, side by side; and if we lay upon a smooth wet plate
four matches, half broken across, a similar attraction brings the
four matches together in the form of a symmetrical cross. Whether
one of these, or yet another, be the explanation of the phenomenon,

* R. S. Lillie, Conditions determining the disposition of the chromatic filaments,
etc., in mitosis; Biol. Bulletin, viii, 1905.

IV] AND STRUCTURE OF THE CELL 315

it is at least plain that by some physical cause, some mutual
attraotion or common repulsion of the particles, we must seek to
account for the symmetry of the so-called "tetrads," and other
more or less familiar configurations. The remarkable annular
chromosomes, shewn in Fig. 95, can be closely imitated by loops
of thread upon a soapy film, when the film within the annulus is
broken or its tension reduced ; the balance of forces is here a simple
one, between the uniform capillary tension which tends to widen out
the ring and the uniform cohesion of its particles which keeps it
together.

We may find other cases, at once simpler and more varied, where
the chromosomes are bodies of rounded form and more or less

Fig. 95. Annular chromosomes, formed in the spermatogenesis of the
mole-cric'iict. From Wilson, after Vom Rath.

uniform size. These also find their way to. an equatorial plate;
we gather (and Lamb assures us) that they are repelled from the
centrosomes. They may go near the equatorial periphery, but they
are not driven there; and we infer that some bond of mutual
attraction holds them together. If they be free to move in a fluid
medium, subject both to some common repulsion and some mutual
attraction, then their circumstances are much like those of Mayer's
well-known experiment of the floating magnets. A number of
magnetised needles stuck in corks, all with like poles upwards, are
set afloat in a basin; they repel one another, and scatter away to
the sides. But bring a strong magnet (of unlike pole) overhead,
and the little magnets gather in under its common attraction, while
still keeping asunder through their own mutual repulsion. The
symmetry of forces leads to a symmetrical configuration, which is

316 ON THE INTERNAL FORM [ch.

the mathematical expression of a physical equiUbrium — and is the
not too remote counterpart of the arrangement of the electrons in
an atom. Be that as it may, it is found that a group of three,
four or five Httle magnets arrange themselves at the corners of an
equilateral triangle, square or pentagon; but a sixth passes within
the ring, and comes to rest in the centre of symmetry of the
pentagon. If there be seven magnets, six form the ring, and the
seventh occupies the centre ; if there be ten, there is a ring of eight
and two within it ; and so on, as follows * :

Number of magnets

5

6

7

8

9

10

11

12

13

14

15

16

Do. in outer ring
Do. in inner ring

5

5

1

6

1

7
1

8

1

8
2

8
3

9
3

10
3

10

4

10
5

11
5

When we choose from the published figures cases where the
chromosomes are as nearly as possible alike in size and form — the
condition necessary for our parallel to hold — then, as LilHe pre-
dicted and as Doncaster and Graham Cannon have shewn, their
congruent arrangement agrees, even to a surprising degree, with
what we are led to expect by theory and analogy (Fig. 96).

The break-up of the nucleus, already referred to and ascribed
to a diminution of its surface-tension, is accompanied by certain
diffusion phenomena which are sometimes visible to the eye; and
we are reminded of Lord Kelvin's view that diffusion is implicitly
associated with surface-tension changes, of which the first step is
a minute puckering of the surface-skin, a sort of interdigitation with
the surrounding medium. For instance, Schewiakoff has observed
in Euglyphaf that, just before the break-up of the nucleus, a system
of rays appears, concentred about it, but having nothing to do with
the polar asters: and during the existence of this striation the
nucleus enlarges very considerably, evidently by imbibition of fluid
from the surrounding protoplasm. In short, diffusion is at work,
hand in hand with, and as it were in opposition to, the surface-
tensions which define the nucleus. By diffusion, hand in hand with
surface-tension, the alveoli of the nuclear meshwork are formed,
enlarged and finally ruptured: diffusion sets up the movements

* H. Graham Cannon, On the nature of the centrosomal force, Journ. Genetics,
XIII, p. 55, 1923.

t Schewiakoff, Ueber die karyokinetische Kerntheilung der Euglypha alveolata,
Morph. Jahrb. xiii, pp. 193-258, 1888 (see p. 216).

IV] AND STRUCTURE OF THE CELL 317

which give rise to the appearance of rays, or striae, around the
nucleus: and through increasing diffusion and weakening surface-
tension the rounded outHne of the nucleus finally disappears.

As we study these manifold phenomena in the individual cases
of particular plants and animals, we recognise a close identity of
type coupled with almost endless variation of specific detail; and

•••

•V.

^»N f" %

o a o 1 ^ ^

(I V J * /•v •

V

# -^

lOj

0.

/•;•;%

Fig. 96. Various numbers of chromosomes in the equatorial plate: the ring-
diagrams give the arrangements predicted by theory. From Graham Cannon.

in particular, the order of succession in which certain of the pheno-
mena occur is variable and irregular. The precise order of the
phenomena, the tiiiie of longitudinal and of transverse fission of
the chromatin thread, of the break-up of the nuclear wall, and so
forth, will depend upon various minor contingencies and ''inter-
ferences." And it is worthy of particular note that these variations
in the order of events and in other subordinate details, while

318 ON THE INTERNAL FORM [ch.

doubtless attributable to specific physical conditions, would seem
to be without any obvious classificatory meaning or other biological
significance.

So far as we have now gone, there is no great difficulty in pointing
to simple and familiar examples of a field of force which are
similar, or comparable, to the phenomena which we witness within
the cell. But among these latter phenomena there are others for
which it is not so easy to suggest, in accordance with known laws,
a simple mode of physical causation. It is not at once obvious
how, in any system of symmetrical forces, the chromosomes, which
had at first been apparently repelled from the poles towards the
equatorial plane, should then be spht asunder, and should presently
be attracted in opposite directions, some to one pole and some to
the other. Remembering that it is not our purpose to assert that
some one particular mode of action is at work, but merely to shew
that there do exist physical forces, or distributions of force, which
are capable of producing the required result, I give the following
suggestive hypothesis, which I owe to my colleague Professor W.
Peddie.

As we have begun by supposing that the nuclear or chromosomal
matter differs in permeability from the medium, that is to say the
cytoplasm, in which it hes, let us now make the further assumption
that its permeabihty is variable, and depends upon the strength of
the field.

In Fig. 97, we have a field of force (representing our cell), con-
sisting of a homogeneous medium, and including two opposite
poles : lines of force are indicated by full lines, and loci of constant
magnitude of force are shewn by dotted lines, these latter being what
are known as Cayley's equipotential curves*.

Let us now consider a body whose permeabihty (/a) depends on
the strength of the field F. At two field -strengths, such as F^, F^,
let the permeability of the body be equal to that of the medium,
and let the curved line in Fig. 98 represent generally its permeabihty
at other field-strengths; and let the outer and inner dotted curves
in Fig. 97 represent respectively the loci of the field-strengths F^,

* Phil. Trans, xiv, p. 142, 1857. Cf. also F. G. Teixeira, TraiU des Courhes,
I, p. 372, Coimbra'. 1908.

IV] AND STRUCTURE OF THE CELL 319

and Fa. The body if it be placed in the medium within either
branch of the inner curve, or outside the outer curve, will tend to
move into the neighbourhood of the adjacent pole. If it be placed

Fb

Fig. 97.

Fig. 98.

in the region intermediate to the two dotted curves, it will tend to
move towards regions of weaker field-strength.

The locus Fly is therefore a locus of stable position, towards which
the body tends to move ; the locus F^ is a locus of unstable position,
from which it tends to move. If the body were placed across F^^,

320 ON THE INTERNAL FORM [ch.

it might be torn asunder into two portions, the split coinciding
with the locus F^.

Suppose a number of such bodies to be scattered throughout the
medium. Let at first the regions ¥„, and F^ be entirely outside the
space where the bodies are situated: and, in making this supposition
we may, if we please, suppose that the loci which we are calhng
Fa and F^, are meanwhile situated somewhat farther from the axis
than in our figure, that (for instance) F^ is situated where we have
drawn F^, and that F^ is still farther out. The bodies then tend
towards the poles; but the tendency may be very small if, in
Fig. 98, the curve and its intersecting straight hne do not diverge
very far from one another beyond Fa\ in other words, if, when

Fig. 99.

situated in this region, the permeability of the bodies is not very
much in excess of that of the medium.

Let the poles now tend to separate farther and farther from one
another, the strength of each pole remaining unaltered; in other
words, let the centrosome-foci recede from one another,- as they
actually do, drawing out the spindle-threads between them. The
loci Fa, F^ will close in to nearer relative distances from the poles.
In doing so, when the locus F^ crosses one of the bodies, the body
may be torn asunder; if the body be of elongated shape, and be
crossed at more points than one, the forces at work will tend to
exaggerate its foldings, and the tendency to rupture is greatest
when Fa is in some median position (Fig. 99).

When the locus Fa has passed entirely over the body, the body
tends to move towards regions of weaker force; but when, in turn,
the locus Fi, has crossed it, then the body again moves towards
regions of stronger force, that is to say, towards the nearest pole.

IV] AND STRUCTURE OF THE CELL 321

And, in thus moving towards the pole, it will do so, as appears
actually to be the case in the dividing cell, along the course of the
outer hues of force, the so-called "mantle-fibres" of the histologist*.

Such considerations as these give general results, easily open to
modification in detail by a change of any of the arbitrary postulates
which have been made for the sake of simpHcity. Doubtless there
are other assumptions which would meet the case; for instance,
that during the active phase of the chromatin molecule (when it de-
composes and sets free nucleic acid) it carries a charge opposite to
that which it bears during its resting, or alkahne phase ; and that it
would accordingly move towards different poles under the influetice
of a current, wandering with its negative charge in an alkahne fluid
during its acid phase to the anode, and to the kathode during its
alkahne phase. A whole field of speculation is opened up when we
begin to consider the cell not merely as a polarised electrical field,
but also as an electrolytic field, full of wandering ions. Indeed it
is high time we reminded ourselves that we have perhaps been
deahng too much with ordinary physical analogies: and that our
whole field of force within the cell is of an order of magnitude where
these grosser analogies may fail to serve us, and might even play
us false, or lead us astray. But our sole object meanwhile, as I
have said more than once, is to demonstrate, by such illustrations
as these, that, whatever be the actual and as yet unknown modus
operandi, there are physical conditions and distributions of force
which could produce just such phenomena of movement as we see
taking place within the living cell. This, and no more, is precisely
what Descartes is said to have claimed for his description of the
human body as a " mechanism f."

While it can scarcely be too often repeated that our enquiry is
not directed towards the solution of physiological problems, save
only in so far as they are inseparable from the problems presented
by the visible configurations of form and structure, and while we
try, as far as possible, to evade, the difficult question of what

* • We have not taken account in the above paragraphs of the obvious fact that
the supposed symmetrical field of force is distorted by the presence in it of the
more or less permeable bodies; nor is it necessary for us to do so, for to that
distorted field the above argument continues to apply, word for word.

t Michael Foster, Lectures on the History of Physiology, 1901, p. 62.

322 ON THE INTERNAL FORM [ch.

particular forces are at work when the mere visible forms produced
are such as to leave this an 6pen question, yet in this particular
case we have been drawn into the use of electrical analogies, and
we are bound to justify, if possible, our resort to this particular
mode of physical action. There is an important paper by R. S. LiUie,
on the "Electrical convection of certain free cells and nuclei*,"
which, while I cannot quote it in direct support of the suggestions
which I have made, yet gives just the evidence we need in order
to shew that electrical forces act upon the constituents of the cell,
and that their action discriminates between the two species of
colloids represented by the cytoplasm and the nuclear chromatin.
And the difference is such that, in the presence of an electrical
current, the cell substance and the nuclei (including sperm-cells)
tend to migrate, the former on the whole with the positive, the
latter with the negative stream : a difference of electrical potential
being thus indicated between the particle and the surrounding
medium, just as in the case of minute suspended particles of various
kinds in various feebly conducting media j*. And the electrical
difference is doubtless greatest, in the case of the cell constituents,
just at the period of mitosis: when the chromatin is invariably
in its most deeply staining, most strongly acid, and therefore,
presumably, in its most electrically negative phase. In short, LilHe
comes easily to the conclusion that "electrical theories of mitosis
are entitled to more careful consideration than they have hitherto
received."

* Amer. J. Physiol, viii, pp. 273-283, 1903 {vide supra, p. 314); cf. ibid, xv,
pp. 46-84, 1905; xxii, p. 106, 1910; xxvii, p. 289, 1911; Journ. Exp. Zool. xv,
p. 23, 1913; etc.

t In like manner Hardy shewed that colloid particles migrate with the negative
stream if the reaction of the surrounding fluid be alkahne, and vice versa. The
whole subject is much wider than these brief allusions suggest, and is essentially
part of Quincke's theory of Electrical Diffusion or Endosmosis: according to
which the particles and the fluid in which they float (or the fluid and the capillary
wall through which it flows) each carry a charge: there being a discontinuity of
potential at the surface of contact and hence a field of force leading to powerful
tangential or shearing stresses, communicating to the particles a velocity which
varies with the density per unit area of the surface charge. See W. B. Hardy's
paper on Coagulation by electricity, Journ. Physiol, xxrv, pp. 288-304, 1899;
also Hardy and H. W. Harvey, Surface electric charges of living cells, Proc. E.S.
(B), Lxxxiv, pp. 217-226, 191 1, and papers quoted therein. Cf. also E. N. Harvey's
observations on the convection of unicellular organisms in an electric field (Studies
on the permeability of cells, Journ. Exp. Zool. x, pp. 508-556, 1911).

IV]

AND STRUCTURE OF THE CELL

323

Among other investigations all leading towards the same general
conclusion, namely that differences of electric potential play their
part in the phenomena of cell division, I would mention a note-
worthy paper by Ida H. Hyde*, in which the writer shews (among
other important observations) that not only is there a measurable
difference of potential between the animal and vegetative poles of
a fertilised egg (Fundulus, toad, turtle, etc.), but also that this
difference fluctuates, or actually reverses its direction, periodically,
at epochs coinciding with successive acts of segmentation or other

Fig. 100. Final stage in the first seg-
mentation of the egg of Cerebra-
tidus. From Prenant, after Coe*.

Fig. 101. Diagram of field of force
with two similar poles.

important phases in the development of the' eggt; just as other
physical rhythms, for instance, in the production of COg , had already
been shewn to do. Hence we need not be surprised to find that th^
"materialised" hues of force, which in the earlier stages form the

* On differences in electrical potential in developing eggs, Amer. Journ. Physiol.
XII, pp. 241-275, 1905. This paper contains an excellent summary, for the time
being, of physical theories of the segmentation of the cell.

t Gray has demonstrated a temporary increase of electrical conductivity in
sea-urchin eggs during the process of fertilisation, and ascribes the changes in
resistance to polarisation of the surface: Electrical conductivity of echinoderm
eggs, etc., Phil. Trans. (B), ccvii, pp. 481-529, 1916.

324

ON THE INTERNAL FORM

[CH.

convergent curves of the spindle, are replaced in the later phases of
caryokinesis by divergent curves, indicating that the two foci, which
are marked out in the field by the divided and reconstituted nuclei,
are now ahke in their polarity* (Figs. 100, 101).

The foregoing account is based on the provisional assumption
that the phenomena of caryokinesis are analogous to those of a
bipolar electrical field — a comparison which seems to offer a helpful
and instructive series of analogies. But there are other forces which
lead to similar configurations. For instance, some of Leduc's
diffusion-experiments offer very remarkable analogies to the dia-
grammatic phenomena of caryokinesis, as shewn in Fig. 102 f.

Fig. 102. Artificial caryokinesis (after Leduc), for comparison with Fig. 88, p. 299.

Here we have two identical (not opposite) poles of osmotic con-
centration, formed by placing a drop of Indian ink in salt water,
and then on either side of this central drop, a hypertonic drop of
salt solution more lightly coloured. On either side the pigment of
the central drop has been drawn towards the focus nearest to it;
but in the middle line, the pigment is drawn in opposite directions
by equal forces, and so tends to remain undisturbed, in the form of
an "equatorial plate."

To account for the same mitotic phenomena an elegant hypothesis
has been put forward by A. B. Lamb J, and developed by Graham

* W. R. Coe, Maturation and fertilisation of the egg of Cerebratulus, Zool.
Jahrbiicher {Anat. Abth.), xii, pp, 425^76, 1899.

t Op. cit. pp. 110 and 91.

t A. B. Lamb, A new explanation of the mechanism of mitosis, Journ. Exp. Zool.
V, pp. 27-33. 1908.

IV] AND STRUCTURE OF THE CELL 325

Cannon*. It depends on certain investigations of the Bjerknes,
father and sonf, which prove that bodies pulsating or oscillating J
in a fluid set up a field of force precisely comparable with the lines
of force in a magnetic field. Certain old and even familiar observa-
tions had pointed towards this phenomenon. Guyot had noticed
that bits of paper were attracted towards a vibrating tuning-fork;
and Schellbach found that a sounding-board so acts on bodies in its
neighbourhood as to attract those which are heavier and repel those
which are lighter than the surrounding medium; in air bits of
paper are attracted and a gas-flame is repelled. To explain these
simple observations, Bjerknes experimented with Httle drums
attached to an automatic bellows. He found that two bodies in
a fluid field, synchronously pulsating or synchronously oscillating,
repel one another when their oscillations are in the same phase, or
their pulsations are in opposite phase; and vice versa: while other
particles, floating passively in the same fluid, tend (as Schellbach
had observed before) to be attracted or repulsed according as they
are heavier or lighter than the fluid medium. The two bodies
behave towards one another like two electrified bodies, or like two
poles of a magnet; we are entitled to speak of them as "hydro-
dynamic poles," we might even call them " hydrodynamic magnets" ;
and pursuing the analogy, we may call the heavy bodies para-
magnetic, and the light ones diamagnetic with regard to them.
Lamb's hypothesis then, and Cannon's, is that the centrosomes act
as "hydrodynamic magnets." The explanation depends on oscilla-
tions which have never been seen, in centrosomes which are not
always to be discovered. But it brings together certain curious
analogies, and these, where we know so little, may be worth
reflecting on.

If we assume that each centrosome is endowed with a vibratory
motion as it floats in the semi-fluid colloids, or hydrosols (to use
Graham's word) of the cell, we ma;^ take it that the visible intra-
cellular phenomena will be much the same as those we have

* Op. cit. Cf. also Gertrud Woken, Zur Physik der Kernteilung, Z. f. allg, Physiol.
XVIII, pp. 39-57, 1918.

t V. Bjerknes, Vorlesungen liber hydrodynamische Fernkrdjte, nach C. A. Bjerknes^
Theorie, Leipzig, 1900.

J A body is said to pulsate when it undergoes a rhythmic change of volume;
it oscillates when it undergoes a rhythmic change of place.

326 ON THE INTERNAL FORM [ch.

described under an electrical hypothesis; the lines of force will
have the same distribution, and such movements as the chromo-
somes undergo, and such symmetrical configurations as they assume,
may be accounted for under the one hypothesis pretty much as
under the other. There are however other phenomena accompanying
mitosis, such as Chambers's astral currents and certain local changes
in the viscosity of the egg, which are more easily explained by the
hydrodynamic theory.

We may assume that the cytoplasm, however complex it may be,
is but a sort of microscopically homogeneous emulsion of high
dispersion, that is to say one in which the minute particles of one
phase are widely scattered throughout, and freely mobile in, the
other; and this indeed is what is meant by caUing it a hydrosol.
Let us assume also that the particles are a little less dense than the
continuous phase in which they are dispersed; and assume lastly
(it is not the easiest of our assumptions) that these ultra-minute
particles will be affected, just as are the grosser ones, by the forces
of the hydrodynamic field.

All this being so, the disperse particles will be repelled from the
oscillating centrosome, with a force which falls off very rapidly, for
Bjerknes tells us that it varies inversely as the seventh power of
the distance; a round clear field, hke a drop or a bubble, will be
formed round the centrosome ; and the disperse particles, expelled
from this region, will tend to accumulate in a crowded spherica-
zone immediately beyond it. Outside of this again they will con-
tinue to be repulsed, but slowly, and we may expect a second and
lesser concentration at the periphery of the cell. A clear central
mass, or "centrosphere," will thus come into being; and the
surrounding cytoplasm will be rendered denser and more viscous,
especially close around the centrosphere and again peripherally, by
condensation of the disperse particles. Moreover, all outward
movements of these lighter particles entail inward movements of
the heavier, which (by hypothesis) are also the more fluid ; stream-
lines or visible currents will flow towards the centre, giving rise to
the star-shaped "aster," and the best accounts of the sea-urchin's
egg* tally well with what is thus deduced from the hydrodynamic

* Cf. R. Chambers, in Journ. Exp. Zool. xxni, p. 483, 1917; Trans. R.S. Canada,
XII, 1918; Journ. Gen. Physiol, ii, 1919.

IV] AND STRUCTURE OF THE CELL 327

hypothesis. The round drop of clear fluid which forms the centre
of the aster grows as the aster grows, fluid streaming towards it
from all parts of the cell along the channels of the astral rays.
The cytoplasni between the rays is in the gel state, but gradually
passes into a sol beyond the confines of the aster. Seifritz asserts
that the substance of the centrosphere is *'not much more viscous
than water," but that the wedges of cytoplasm between the inwardly
directed streams are stiff and viscous*.

After the centrosome divides we have two oscillating bodies
instead of one; they tend to repel one another, and pass easily
through the fluid centrosphere to the denser layer around. But
now the new centrosomes, on opposite sides of the centrosphere,
repel, each on its own side, the disperse particles of the denser zone ;
and two new asters are formed, their rays marked by the streams
coursing inwards to the centrosome-foci. Thus the amphiaster
comes into being; it is not that the old aster divides, as a definite
entity; but the old aster ceases to exist when its focus is disturbed,
and about the new foci new asters are necessarily and automatically
developed. Again this hypothetic account taUies well with Chambers's
description.

The same attractions and repulsions should be manifested, perhaps
better still, in whatsoever bodies he or float within the cell, whether
liquid or solid, oil-globules, yolk-particlea, mitochondria, chromo-
somes or what not. A zoned, concentric arrangement of yolk-
globules is often seen in the egg, with the centrosome as focus;
and in certain sea-urchin eggs the mitochondria gather around
the centrosome while the amphiaster is forming, collecting together
in that very zone to which Chambers ascribes a semi-rigid or viscous
consistency!. The Golgi bodies found in various germ-cells are at
first black rod-hke bodies embedded in the centrosphere ; they
undergo changes and complex movements, now scattering through
the cytoplasm and anon crowding again around the centrosome.
Some periodic change in the density of these bodies compared with

♦ Cf. W. Seifritz, Some physical properties of protoplasm, Ann. Bot. xxxv,
1921. Wo. Ostwald and M. H. Fischer had thought that the astral rays were
due to local changes of the plasma-sol into a gel, Zur physikal. chem. Theorie der
Befruchtung, Pfluger's Archiv, cvi, pp. 2^3-266, 1905.

t Cf. F. Vejdovsky and A. Mrazek, Umbildung des Cytoplasma wahrend der
Befruchtung und Zelltheilung, Arch. f. mikr. Anat. LXii. 431-579, 1903.

328 ON THE INTERNAL FORM . [ch.

that of the medium in which they He seems all that is required
to account for their excursions; and such changes of density are
not only of likely occurrence during the active chemical operations
associated with fertilisation and division, but are in all probability
inseparable from the changes in viscosity which are known to
occur*. The movements and arrangements of the chromosomes,
already described, may be easily accounted for if we postulate, in
addition to their repulsion from the oscillating centrosomes, induced
oscillations in themselves such as to cause them to attract one another.

The well-defined length of the spindle and the position of equili-
brium in which it comes to rest may be conceived as resultants of
the several mutual repulsions of the centrosomes by one another,
by the chromosomes or other lighter material of the equatorial plate,
and again by such lighter material as may have accumulated at the
periphery of the egg ; the first two of these will tend to lengthen the
spindle, the last to shorten it; and the last will especially affect its
position and direction. When Chambers amputated part of an
amphiastral egg, the remains of the amphiaster disappeared, and
then came into being again in a new and more symmetrical position ;
it or its centrosomal focus had been symmetrically repelled, we may
suppose, by the fresh surface. Hertwig's law that the spindle-axis
tends to lie in the direction of the largest mass of protoplasm, in
other words to point where the cell-surface lies farthest off and its
repulsion is least felt, may likewise find its easy explanation.

Between these hypotheses we may choose one or other (if we
choose at all), according to our judgment. As Henri Poincare tells
us, we never know that any one physical hypothesis is true, we take
the simplest we can find; and this we call the guiding principle of
simphcity ! In this case, the hydrodynamic hypothesis is a simple
one; but it all rests on a hypothetic oscillation of the centrosomes,
which has never been witnessed. Bayliss has shewn that precisely
such reversible states of gelation as we have been speaking of as

* Cf. G. Odquist, Viscositatsanderungen des Zellplasmas wahrend der ersten
Entwicklungsstufen des Froscheies, Arch. f. Entw. Mech. ia, pp. 610-624, 1922;
A. Gurwitsch, Pramissen und anstossgebende Faktoren der Furchung und
Zelltheilung, Arch. f. Zellforsch. n, pp. 495-548, 1909; L. V. Heilbrunn,
Protoplasmic viscosity-changes during mitosis, Journ. Exp. Zool. xxxiv, pp. 417-447,
1921 ; ibid, xliv, pp. 255-278, 1926; E. Leblond, Passage de I'etat de gel a I'etat de sol
dans le protoplasme vivant, C.R. Soc. Biol, lxxxii, p. 1150; of. ibid. p. 1220; etc.

rv] AND STRUCTURE OF THE CELL 329

"periodic changes in viscosity" may be induced in living protoplasm
by electrical stimulation*. On the other hand, the fact that the
hydrodynamic forces fall off as fast as they do with increasing
distance Hmits their efficacy ; and the minute disperse particles
must, under Stokes's law, be slow to move. Lastly, it may well be
(as Lillie has urged) that such work as his own, or Ida Hyde's, or
Gray's, on change of potential in developing eggs, taken together
with that of many others on the behaviour of colloid particles in an
electrical field, has not yet been followed out in all its consequences,
either on the physical or the physiological side of the problem.

But to return to our general discussion.

As regards the actual mechanical division of the cell into two
halves, we shall see presently that, in certain cases, such as that
of a long cylindrical filament, surface-tension, and what is known
as the principle of "minimal areas," go a long way to explain the
mechanical process of division; and in all cells whatsoever, the
process of division must somehow be explained as the result of a
conflict between surface-tension and its opposing forces. But in
such a case as our spherical cell, it is none too easy to see what
physical cause is at work to disturb its equiUbrium and its integrity.

The fact that when actual division of the cell takes place, it does
so at right angles to the polar axis and precisely in the direction
of the equatorial plane, would lead us to suspect that the new
surface formed in the equatorial plane sets up an annular tension,
directed inwards, where it meets the outer surface layer of the cell
itself. But at this point the problem becomes more comphcated.
Before we can hope to comprehend it, we shall have not only to
enquire into the potential distribution at the surface of the cell in
relation to that which we have seen to exist in its interior, but also
to take account of the differences of potential which the material
arrangements along the lines of force must themselves tend to
produce. Only thus can we approach a comprehension of the
balance of forces which cohesion, friction, capillarity and electrical
distribution combine to set up.

The manner in which we regard the phenomenon would seem to

* W. M. Bayliss, Reversible gelation in living protoplasm, Proc. R.S. (B), xci,
pp. 196-201, 1920.

PO ON THE INTERNAL FORM [ch.

turn, in great measure, upon whether or no we are justified in
assuming that, in the hquid surface-film of a minute spherical cell,
local and symmetrically localised differences of surface-tension are
likely to occur. If not, then changes in the conformation of the
cell such as lead immediately to its division must be ascribed not
to local changes in its surface-tension, but rather to direct changes
in internal pressure, or to mechanical forces due to an induced
surface-distribution of electrical potential. We have little reason to
be sceptical ; in fact we now know that the cell is so far from being
chemically and physically homogeneous that local variations in its
surface-tension are more than likely, they are certain to occur.

Biitschh suggested more than sixty years ago that cell-division
was brought about by an increase of surface-tension in the equatorial
region of the cell ; and the suggestion was the more remarkable that
it was (I beheve) the very first attempt to invoke surface-tension
as a factor in the physical causation of a biological phenomenon*.
An increase of equatorial tension would cause the surface-area there
to diminish, and the equator to be pinched in, but the total surface-
area of the cell would be increased thereby, and the two effects
would strike a balance f. But, as Biitschh knew very well, the
surface-tension change would not stand alone; it would bring other
phenomena in its train, currents would tend to be set up, and
tangential strains would be imposed on the cell-membrane or cell-
surface as a whole. The secondary if not the direct effects of
increased equatorial tension might, after all, suffice for the division of
the cell. It was Loeb, in 1895, who first shewed that streaming went
on from the equator towards the divided nuclei. To the violence
of these streaming movements he attributed the phenomenon of
division, and many other physiologists have adopted this hypo-
thesis J. The currents of which Loeb spoke call for counter-currents

* 0. Butschli, tJber die ersten Entwicklungsvorgange der Eizelle, Abh.
Senckenberg. naturf. Gesellsch. x, 1876; Uber Plasmastrqmungen bei der Zell-
theilung, Arch. f. Entw. Mech. x, p. 52, 1900. Ryder ascribed the earyokinetic
figures to surface-tension in his Dynamics in Evolution, 1894.

t A relative, not positive, increase of surface-tension, was part of Giardina's
hypothesis: Note sul mecanismo della divisione cellulare, Anat. Anz. xxi. 1902.

J J. Loeb, Amer. Journ. Physiol, vi, p. 432, 1902; E. G. Conklin, Protoplasmic
movements as a factor in differentiation, Wood's Hole Biol. Lectures, p. 69, etc.,
1898-99; J. Spek, Oberflachenspannungsdifferenzen als eine Ursache der Zell-
teilung, Arch.f. Entw. Mech. xliv, pp. 54-73, 1918.

IV] AND STRUCTURE OF THE CELL 331

towards the equator, in or near the surface of the cell; and theory
and observation both indicate that precisely such currents are bound
to be set up by the surface-energy involved in the increase of
equatorial tension.

An opposite view has been held by some, and especially by
T. B. Robertson*. Quincke had shewn that the formation of soap
at the surface of an oil-droplet lowers the surface-tension of the
latter, and that if the saponification be local, that part of the surface
tends to enlarge and spread out accordingly. Robertson, in a very
curious experiment, found that by laying a thread, moistened with
dilute caustic alkali or merely smeared with soap, across a drop of
olive oil afloat in water, the drop at once divided into two. A
vast amount of controversy has arisen over this experiment, but
Spek seems to have shewn conclusively that it is an exceptional
case.

In a drop of olive-oil, balanced in water f and touched anywhere
with an alkaH, there is so copious a formation of lighter soaps that
di^erences of density tend to drag the drop in two. But in the
case of other oils (and especially the thinner oils, such as oil of
bergamot) the saponified portion bulges, as theory directs; and
when the alkali is applied to two opposite poles the equatorial
region is pinched in, as McClendonJ, in opposition to Robertson,
had found it to do. Conversely, if an alkaline thread be looped
around the drop, the zone of contact bulges, and instead of dividing
at the equator the drop assumes a lens-like form.

We may take it then as proven that a relative increase of equatorial
surface-tension, whether in oil-drops, mercury-globules or living
cells, does lead, or tend to lead, to an equatorial constriction. In
all cases a system of surface-currents is set up among the fluid drops
towards the zone of increased tension ; and an axial counter-current
flows towards the pole or poles of lowered tension. Precisely such
currents have been observed to run in various eggs (especially of

♦ T. B. Robertson, Note on the chemical mechanics of cell-division, Arch. f.
Entw. Mech. xxvn, p. 29, 1909; xxxii, p. 308, 1911; xxxv, p. 402, 1913. Cf.
R. S. Lillie, Joum.-Exp. Zool. xxi, pp. 369^02, 1916; McClendon, loc. cit.; etc.

t In these experiments, and in many of Quincke's, a little chloroform is added
to the oil, in order to bring its density as near as may be to that of water.

X J. F. McClendon, Note on the mechanics of cell -division, Arch. f. Entw. Mech.
xxxrv, pp. 263-266, 1912.

332 ON THE INTERNAL FORM [ch.

certain Nematodes) during division of the cell; but if the })rocess
be slow, more than 7 or 8 minutes long, the slow currents become
hard to see. Various contents of the cell are transported by these
currents, and clear, yolk-free polar caps and equatorial accumula-
tions of yolk and pigment are among the various manifestations of
the phenomenon. The extrusion of a polar body, at a small and
sharply defined region of lowered tension, is a particular case of the
same principle*.

But purely chemical changes are not of necessity the fundamental
cause of alteration in the surface-tension of the egg, for the action
of electrolytes on surface-tension is now well known and easily
demonstrated. So, according to other views than those with which
we have been dealing, electrical charges are sufficient in themselves
to account for alterations of surface-tension, and in turn for that
protoplasmic streaming which, as so many investigators agree,
initiates the segmentation of the eggf. A great part of our difficulty
arises from the fact that in such a case as this the various pheno-
mena are so entangled and apparently concurrent that it is hard
to say which initiates another, and to which this or that secondary
phenomenon may be considered due. Of recent years the pheno-
menon of adsorption has been adduced (as we have already briefly
said) in order to account for many of the events and appearances
which are associated with the asymmetry, and lead towards the
division, of the cell. But our short discussion of this phenomenon
may be reserved for another chapter.

However, we are not directly concerned here with the phenomena
of segmentation or cell-division in themselves, except only in so far
as visible changes of form are capable of easy and obvious correla-
tion with the play of force. The very fact of "development"
indicates that, while it lasts, the equihbrium of the egg is never
complete J. And the gist of the matter is that, if you have caryo-
kinetic figures developing inside the cell, that of itself indicates that
the dynamic system and the localised forces arising from it are in

* J. Spek, loc. cit. pp. 108-109.

t Cf. D'Arsonval, Relation entre la tension superficielle et certains phenomenes
electriques d'origine animale. Arch, de Physiol, i, pp. 460-472, 1889; Ida H. Hyde,
op. cit. p. 242.

I Cf. Plateau's remarks (Statique des liquides, ii, p. 154) on the tendency towards
equilibrium, rather than actual equilibrium, in many of his systems of soap-films.

IV] AND STRUCTURE OF THE CELL 333

gradual alteration; and changes in the outward configuration of
the system are bound, consequently, to take place.

Perhaps we may simplify the case still more. We have learned
many things about cell-division, but we do not know much in the
end. We have dealt, perhaps, with too many related phenomena,
and failed because we tried to combine and account for them all.
A physical problem, still more a mathematical one, wants reducing
to its simplest terms, and Dr Rashevsky has simplified and general-
ised the problem of cell-division (or division of a drop) in a series of
papers, which still outrun by far the elementary mathematics of
this book. If we cannot follow^ him in all he does, we may find
useful lessons in his way of doing it. Cells are of many kinds; they
differ in size and shape, in visible structure and chemical com-
position. Most have a nucleus, some few have none; most need
oxygen, some few do not; some metabolise in one way, some in
another. What small residuum of properties remains common to
them all? A living cell is a little fluid (or semi-fluid) system, in
which work is being done, physical forces are in operation and
chemical changes are going on. It is in such intimate relation with
the world outside — its own milieu interne with the great ynilieu
externe — that substances are continually entering the cell, some to
remain there and contribute to its growth, some to pass out again
with loss of energy and metabolic change. The picture seems
simplicity itself, but it is less simple than it looks. For on either
side of the boundary- wall, both in the adjacent medium and in the
living protoplasm within, there will be no uniformity, but only
degrees of activity, and gradients of concentration. Substances
which are being absorbed and consumed will diminish from periphery
to centre; those which are diffusing outwards have their greatest
concentration near the centre, decrease towards the periphery, and
diminish further with increasing distance in the near neighbourhood of
the system. Size, shape, diffusibility, permeability, chemical properties
of this and that, may affect the gradients, but in the living cell the
interchanges are always going on, and the gradients are always there *.

* Outward diffusion makes one of the many contrasts between cell-growth and
crystal-growth. But the diffusion^gradients round a growing crystal are far more
complicated than was once supposed. Cf. W. F. Berg, Crystal growth from
solutions, Proc. R.8. (A), clxiv, pp. 79-95, 1938.

334 ON THE INTERNAL FORM [ch.

If the cell be homogeneous, taking in and giving out at a constant
rate in a uniform way, its shape will be spherical, the concentration-
field of force, or concentration-field, will likewise have a spherical
symmetry, and the resultant force will be zero. But if the symmetry
be ever so little disturbed, and the shape be ever so little deformed,
then there will be forces at work tending to increase the deformation,
and others tending to equalise the surface-tension and restore the
spherical symmetry, and it can be shewn that such agencies are
within the range of the chemistry of the- cell. Since surface-
tension becomes more and more potent as the size of the drop
diminishes, it follows that (under fluid conditions) the smallest
solitary cells are least likely to depart from a spherical shape, and
that cell-division is only likely to occur in cells above a certain
critical order of magnitude; and using such physical constants
as are available, Rashevsky finds that this critical magnitude
tallies fairly well with the average size of a living cell. The more
important lesson to learn, however, is this, that, merely by virtue
of its metabolism, every cell contains within itself factors which may
lead to its division after it reaches a certain critical size.

There are simple corollaries to this simple setting of the case.
Since unequal concentration-gradients are the chief cause which
renders non-spherical shapes of cell possible, and these last only so
long as the cell lives and metabolises, it follows that, as soon as the
gradients disappear, whether in death or in a "resting-stage", the
cell reverts to a spherical shape and symmetry. Again, not only is
there a critical size above which cell-division becomes possible,,
and more and more probable, but there must also be a size beyond
which the cell is not likely to grow. For the "specific surface"
decreases, the metabolic exchanges diminish, the gradients become
less steep, and the rate of growth decreases too ; there must come
a stage where anabolism just balances katabolism, and growth
ceases though life goes on. When streaming currents are visible
within the cell, they seem to complicate the problem; but after all,
they are part of the result, and proof of the existence, of the gradients
•we have described. In any further account of Rashevsky 's theories
the mathematical difficulties very soon begin. But it is well to
realise that pure theory often carries the mathematical physicist a
long way ; and that higher and higher powers of the microscope, and

IV] AND STRUCTURE OF THE CELL 335

greater and greater histological skill are not the one and only way
to study the physical forces acting within the cell *.

As regards the phenomena of fertihsation, of the union of the
spermatozoon with the "pronucleus" of the egg, we might study
these also in illustration, up to a certain point, of the forces which
are more or less manifestly at work. But we shall merely take, as
a single illustration, the paths of the male and female pronuclei, as
they travel to their ultimate meeting-place.

The spermatozoon, when within a very short distance of the egg-
cell, is attracted by it, the same attraction being further manifested
in a small conical uprising of the surface of the eggt. The nature
of the attractive force has been much disputed. Loeb found the
spermatozoon to be equally attracted by other substances, even by
a bead of glass. It has been held also that the attraction is
chemotropic, some substance being secreted by the egg which drew
the sperm towards it: just as Pfeifer, having shewn that maUc acid
has an attraction for fern-antheridia, supposed this substance to
play its attractive part within the mucus of the archegonia. Again,
the chemical secretion may be neither attractive nor directive, but
yet play a useful part in activating the spermatozoa. However
that may be, Gray has shewn reason to believe that an electromotive
force is developed in the contact between active spermatozoon and
inactive ovum; and that it is the electrical change so set up, and
almost instantaneously propagated, which precludes the entry of
another spermatozoon J. Whatever the force may be, it is one
which acts normally to the surface of the ovum, and after entry the

* Cf. N. Rashevsky, Mathematical Biophysics, Chicago, 1938; and many earlier
papers. Eg. Physico-matheraatical aspects of cellular multiplication and de-
velopment, Cold Spring Harbor Symposia, ii, 1934; The mechanism of division of
small liquid systems which are the seat of physico-chemical reactions, Physics, in,
pp. 374-379, 1934; papers in Protoplasma, xiv-xx, 1931-33, etc.

t With the classical account by H. Fol, C.R. lxxxiii, p. 667, 1876; Mem. Soc.
Phys. Geneve, xxvi, p. 89, 1879, cf. Robert Chambers, The mechanism of the entrance
of sperm into the star-fish egg, Journ. Gen, Physiol, v, pp. 821-829, 1923. Here
a delicate filament is said to run out from the fertilisation -cone and drag the
spermatozoon in; but this is disputed and denied by E. Just, Biol. Bull. LVii,
pp. 311-325, 1929.

I But, under artificial conditions, "polyspermy" may take place, eg. under
the action of dilute poisons, or of an abnormally high temperature, these being
doubtless also conditions under which the surface-tension is diminished.

336 ON THE INTERNAL FORM [ch.

spermatozoon points straight towards the centre of the egg. From
the fact that other spermatozoa, subsequent to the first, fail to
effect an entry, we may safely conclude that an immediate con-
sequence of the entry of the spermatozoon is an increase in the
surface-tension of the egg: this being but one of the complex
reactions exhibited by the surface, or cortex of the cell*. Some-
where or other, within the egg, near or far away, lies its own nuclear
body, the so-called female pronucleus, and we find that after a
while this has fused with the "male pronucleus" or head of the
spermatozoon, and that, the body resulting from their fusion has
come to occupy the centre of the egg. This must be due (as Whitman
pointed out many years ago) to a force of attraction acting between
the two bodies, and another force acting upon one or other or both
in the direction of the centre of the cell. Did we know the magnitude
of these several forces, it would be an easy task to calculate the
precise path which the two pronuclei would follow, leading to con-
jugation and to the central position. As we do not know the
magnitude, but only the direction, of these forces, we can only make
a general statement: (1) the paths of both moving bodies will he
wholly within a plane triangle drawn between the two bodies and
the centre of the cell; (2) unless the two bodies happen to he, to
begin with, precisely on a diameter of the cell, their paths until they
meet one another will be curved paths, the convexity of the curve
being towards the straight line joining the two bodies; (3) the two
bodies will meet a httle before they reach the centre; and, having
met and fused, will travel on to reach the centre in a straight hne.
The actual study and observation of the path followed is not very
easy, owing to the fact that what we usually see is not the path
itself, but only a projection of the path upon the plane of the
microscope ; but the curved path is particularly well seen in the frog's
egg, where the path of the spermatozoon is marked by a little streak
of brown pigment, and the fact of the meeting of the pronuclei before
reaching the centre has been repeatedly seen by many observers f.

* See Mrs Andrews' beautiful observations on "Some spinning activities of
protoplasm in starfish and echinoid eggs," J own. Morphol. xii, pp. 307-389, 1897.

t W. Pfeffer, Locomotorische Richtungsbewegungen durch chemische Reize,
Unters. a. d. Botan. Inst. Tubingen, i, 1884; Physiology of Plants, m, p. 345, Oxford,
1906; W. J. Dakin and M. G. C, Fordham, Journ. Exp. Biol. t. pp. 183-200, 1924.
Cf. J. Loeb, Dynamics of Living Matter, 1906, p. 153.

IV] AND STRUCTURE OF THE CELL 337

The problem recalls the famous problem of three bodies, which has
so occupied the astronomers; and it is obvious that the foregoing
brief description is very far from including all possible cases.
Many of these are particularly described in the works of Fol, Roux,
Whitman and others*.

The intracellular phenomena of which we have now spoken have
assumed great importance in biological literature and discussion
during the last fifty years; but it is open to us to doubt whether
they will be found in the end to possess more than a secondary,
even a remote, biological significance. Most, if not all of them,
would seem to follow immediately and inevitably from certain
simple assumptions as to the physical constitution of the cell, and
from an extremely simple distribution of polarised forces within it.
We have already seen that how a thing grows, and what it grows
into, is a dynamic and not a merely material problem; so far as
the material substance is concerned, it is so only by reason of the
chemical, electrical or other forces which are associated with it.
But there is another consideration which would lead us to suspect
that many features in the structure and configuration of the cell
are of secondary biological importance; and that is, the great
variation to which these phenomena are subject in similar or closely
related organisms, and the apparent impossibihty of correlating
them with the pecuHarities of the organism as a whole. In a
broad and general way the phenomena are always the same. Certain
structures swell and contract, twine and untwine, split and unite,
advance and retire ; certain chemical changes also repeat themselves.
But Nature rings the changes on all the details. "Comparative
study has shewn that almost every detail of the processes (of
mitosis) described above is subject to variation in different forms
of cells |." A multitude of cells divide to the accompaniment of
caryokinetic phenomena; but others do so without any visible
caryokinesis at all. Sometimes the polarised field of force is within,

* H. Fol, Becherches sur la fecondation, 1879; W. Roux, Beitrage zur Erit-
wickelungsmechanik des Embryos, Arch. f. Mikr. Anat. xix, 1887; C. 0. Whitman,
Ookinesis, Journ. Morph. i, 1887; E. Giglio-Tos, Entwicklungsmechanische
Studien, I, Arch. f. Entw. Mech. li, p. 94, 1922, See also Frank R. Lillie, Problems
of Fertilisation, Chicago, 1919.

t Wilson, The Cell. p. 77; cf. 3rd ed. (1925), p. 120.

338 FORM AND STRUCTURE OF THE CELL [ch.

sometimes it is adjacent to, and at other times it lies remote from,
the nucleus. The distribution of potential is very often symmetrical
and bipolar, as in the case described; but a less symmetrical
distribution often occurs, with the result that we have, for a time
at least, numerous centres of force, instead of the two main correlated
poles : this is the simple explanation of the numerous stellate figures,

Haploid number of chr.'raosomes
Fig. 103. Summation diagram shewing the % number of instances (among 2,415
phanerogams and 1,070 metazoa), in which the chromosomes do not exceed
a given number. Data from M. J. D. White.

or "Strahlungen," which have been described in certain eggs, such as
those of Chaetopterus. The number of chromosomes may be constant
within a group, as in the tailed Amphibia, with 12; or very variable,
as in sedges, and in grasshoppers * ; in one and the same species
of worm (Ascaris megdlocephala), one group or two groups of
chromosomes may be present. And remarkably constant, in
general, as the number in any one species undoubtedly is, yet we
must not forget that, in plants and animals alike, the whole range
of observed numbers is but a small one (Fig. 103); for (as regards

* There are varieties of Artemia salina which hardly differ in outward characters,
but differ widely in the number of their chromosomes.

IV] OF CHROMOSOMES 339

the germ-nuclei) few have less than six chromosomes, and few have
more than twenty*. In closely related animals, such as various
species of Copepods, and even in the same species of worm or insect,
the form of the chromosomes and their arrangement in relation* to
the nuclear spindle have been found to differ in ways alluded to
above ; while only here and there, as among the chrysanthemums,
do related species or varieties shew their own characteristic chromo-
some numbers. In contrast to the narrow range of the chromo-
some numbers, we may reflect on the all but infinite possibilities of
chemical variabihty. Miescher shewed that a molecule containing
40 C-atoms would admit (arithmetically though not necessarily
chemically) of a million possible isomers; and changes in position
of the N-atoms of a protein, for instance, might vastly increase
that prodigious number. In short, we cannot help perceiving
that many nuclear phenomena are not specifically related to the
particular organism in which they have been observed, and that
some are not even specially and indisputably connected with the
organism as such. They include such manifestations of the physical
forces, in their various permutations and combinations, as may also
be witnessed, under appropriate conditions, in non-living things.
When we attempt to separate our purely morphological or "purely

* The commonest numbers of (haploid) chromosomes, both in plants and
animals, are 8, 12 and 16. The median number is 12 in both, and the lower
quartile is 8, likewise in both; but the upper quartile is 24 or thereby in animals,
and in the neighbourhood of 16 in plants. If we may judge by the long lists given
by E. B. Wilson (The Cell, 3rd ed. pp. 855-865), by M. Ishikawa in Botan. Mag.
Tokyo, XXX, 1916, by M. J. D. White in his book on Chromosomes, or by Tischler
in Tabulae Biologicae (1927), fully 60 per cent, of the observed cases lie between 6
and 16. As Wilson says (p. 866) "the number of chromosomes is per se a matter
of secondary importance"; and (p. 868) "We must admit the present inadequacy
of attempts to reduce the chromosome numbers to any single or consistent
arithmetical rules." Clifford Dobell had said the same thing: "Nobody nowadays
will be prepared to argue that chromosome numbers, as such, have any quantitative
or qualitative relation to the characters exhibited by their owners. Complexity
of bodily structure is certainly not correlated in any way with multiplicity of
chromosomes " ; La Cellule, xxxv, p. 188, 1924. On the other hand, Tischler stoutly
maintains that chromosome-numbers give useful evidence of phylogenetic affinity
{Biol. Centralbl. XLvni, pp. 321-345, 1928); and there axe a few well-known cases,
such as the chrysanthemums, where, undoubtedly, the numbers are constant and
specific. Again in certain cases, the number of the chromosomes may- differ in
dififerent races (diploid and tetraploid) of the same plant; and the difference is
accompanied by differences in cell-size, in rate of growth, and even in the shape
of the fruit (of. Sinnott and Blakeslee, Xat. Acad, of Sci. 1938, p. 476).

340 FORM AND STRUCTURE OF THE CELL [ch.

embryological " studies from physiological and physical investiga-
tions, we tend ipso facto to regard each particular structure and
configuration as an attribute, or a particular "character," of this or
that particular organism. From this assumption we are easily led to
the framing of theories as to the ancestral history, the classificatory
position, the natural affinities of the several organisms : in fact, to
apply our embryological knowledge to the study of phylogeny.
When we find, as we are not long of finding, that our phylogenetic
hypotheses become complex and unwieldy, we are nevertheless
reluctant to admit that the whole method, with its fundamental
postulates, is at fault; and yet nothing short of this would seem
to be the case, in regard to the earher phases at least of embryonic
development. All the evidence at hand goes, as it seems to me, to
shew that embryological data, prior to and even long after the
epoch of segmentation, are essentially a subject for physiological and
physical investigation and have but the shghtest fink, if any, with
the problems of zoological classification. Comparative embryology
has its own facts to classify, and its own methods and principles of
classification. We may classify eggs according to the presence or
absence, the paucity or abundance, of their associated food-yolk,
the chromosomes according to their form and their number, the
segmentation according to its various "types" — radial, bilateral,
spiral, and so forth. But we have httle right to expect, and in
point of fact we shall very seldom and (as it were) only accidentally
find, that these embryological categories coincide with the lines of
"natural" or "phylogenetic" classification which have been arrived
at by the systematic zoologist.

The efforts to explain "heredity" by help of "genes" and chromo-
somes, which have grown up in the hands of Morgan and others since
this book was first written, stand by themselves in a category which
is all their own and constitutes a science which is justified of
itself. To weigh or criticise these explanations would lie outside
my purpose, even were I fitted to attempt the task. When these
great discoveries began to be made, Bateson crossed the ocean
to see and hear for himself what Morgan and his pupils had to
shew and to tell. He came home convinced, and humbly marvelling.
And I leave this great subject on one side not because I doubt for a
moment the facts nor dispute the hypotheses nor decry the im-

IV] OF THE CELL-THEORY 341

portance of one or other; but because we are so much in the dark
as to the mysterious field of force in which the chromosomes he,
far from the visible horizon of physical science, that the matter lies
(for the present) beyond the range of problems which this book
professes to discuss, and the trend of reasoning which it endeavours
to mamtain.

The cell*, which Goodsir spoke of as a centre of force, is in reality
a sphere of action of certain more or less localised forces; and of
these, surface-tension is the particular force which is especially
responsible for giving to the cell its outline and its morphological
individuahty. The partially segmented differs from the totally
segmented egg, the unicellular Infusorian from the minute multi-

* The " cell -theory " began early and grew slowly. In a curious passage which
Mr Clifford Dobell has shewn me {Nov. Org. ii, 7, ad fin.). Bacon speaks of "cells"
in the human body: of a " coUocatio spiritus per corpoream molem, eiusque pori,
meatus, venae et cellulae, et rudimenta sive tentamenta corporis organici." It is
" surely one of the most strangely prophetic utterances which even Bacon ever
made." Apart from this the story begins in the seventeenth century, with Robert
Hooke's well-known figure of the "cells" in a piece of cork (1665), with Grew's
"bladders" or "bubbles" in the parenchyma of young beans, and Malpighi's
"utriculi" or "sacculi" in the parenchyma or "utriculorum substantia" of
various plants. Christian Fr. v. Wolff conceived, about the same time, a hypo-
thetical "cell-theory," on the analogy of Leibniz's Monads; but the first clear
idea of a cellular parenchyma, or contextus cellularis, came from C. Gottlieb
Ludwig (1742), and from K. Fr. Wolff, who spoke freely of cells or cellulae.
Fontana, author of a curious Traite sur le venin de la vipere (1781), described
various histological elements, caught a glimpse of the nucleus, and experi-
mented with reagents, using syrup of violets for a stain. Early in the
eighteenth century the vessels of the plant played an important role, under Kurt
Sprengel and Treviranus; but it was not till 1831 that Hugo v. Mohl recognised
that they also arose from "cells." About this time Robert Brown discovered,
or re-discovered, the nucleus (1833), which Schleiden called the cytohlast, or "cell-
producer." It was Schleiden's idea, and a far-seeing one, that the cell lived a double
life, a life of its own and the life of the plant to which it belonged: "jede Zelle
fuhrt nun ein zweifaches Leben : ein selbststandiges, nur ihrer eigenen Entwicklung
angehorigen, und ein anderes mittelbares, insofern sie integrierender Theil einer
Pfianze geworden ist " {Phylogenesis, 1838, p. 1 ). The cell-theory, so long a- building,
may be said to have been launched, and christened, with Schwann's Mikroskopische
Untersmhungen of 1839. Within the next five years Martin Barry shewed how
cell-division starts with the nucleus, Henle described the budding of certain cells,
and Goodsir declared that all cells originate in pre-existing cells, a doctrine at once
accepted by Remak, and made famous in pathology by Virchow. (Cf. {int. al.)
J. G. McKendrick, On the modern cell-theory, etc., Proc Phil. Soc. Glasgow, xix,
pp. 1-55, 1887; J. Stephenson, Robert Brown. . .and the cell-theory, Proc. Linn.
Soc. 1931-2, pp. 45-54; M. Mobius, Hundert Jahre Zellenlehre, Jen. Ztschr. lxxi,
pp. 313-326, 1938.)

342 FORM AND STRUCTURE OF THE CELL [ch.

cellular Turbellarian, in the intensity and the range of those surface-
tensions which in the one case succeed and in the other fail to form
a visible separation between the cells. Adam Sedgwick used to
call attention to the fact that very often, even in eggs that appear
to be totally segmented, it is yet impossible to discover an actual
separation or cleavage, through and through, between the cells which
on the surface of the egg are so clearly delimited; so far and no
farther have the physical forces effectuated a visible "cleavage."
The vacuolation of the protoplasm in Actinophrys or Actinosphaerium
is due to localised surface-tensions, quite irrespective of the multi-
nuclear nature of the latter organism. In short, the boundary walls
due to surface-tension may be present or may be absent, with or
without the delimination of the other specific fields of force which
are usually correlated with these boundaries and with the inde-
pendent individuality of the cells. What we may safely admit,
however, is that one effect of these circumscribed fields of force is
usually such a separation or segregation of the protoplasmic
constituents, the more fluid from the less fluid and so forth, as to
give a field where surface-tension may do its work and bring a
visible bouhdary into being. When the formation of a "surface"
is once effected, its physical condition, or phase, will be bound to
differ notably from that of the interior of the cell, and under
appropriate chemical conditions the formation of an actual cell- wall,
cellulose or other, is easily inteUigible. To this subject we shall
return again, in another chapter.

From the moment that we enter on a dynamical conception of
the cell, we perceive that the old debates were vain as to what
visible portions of the cell were active or passive, living or non-
living. For the manifestations of force can only be due to the
interaction of the various parts, to the transference of energy from
one to another. Certain properties may be manifested, certain
functions may be carried on, by the protoplasm apart from the
nucleus; but the interaction of the two is necessary, that other
and more important properties or functions may be manifested.
We know, for instance, that portions of an Infusorian are incapable
of regenerating lost parts in the absence of a nucleus, while nucleated
pieces soon regain the specific form of the organism: and we are
told that reproduction by fission cannot be initiated, though

IV] OF THE CELL-THEORY 343

apparently all its later steps can be carried on, independently of
nuclear action. Nor, as Verworn pointed out, can the nucleus
possibly be regarded as the "sole vehicle of inheritance," since only
in the conjunction of cell and nucleus do we find the essentials of
cell-hfe. "Kern und Protoplasma sind nur vereint lebensfahig," as
Nussbaum said. Indeed we may, with E. B. Wilson, go further,
and say that "the terms 'nucleus' and 'cell-body' should probably
be regarded as only topographical expressions denoting two
differentiated areas in a common structural basis."

Endless discussion has taken place regarding the centrosome,
some holding that it is a specific and essential structure, a permanent
corpuscle derived from a similar pre-existing corpuscle, a "fertilising
element" in the spermatozoon, a special "organ of cell-division,"
a material "dynamic centre" of the cell (as Van Beneden and
Boveri call it); while on the other hand, it is pointed out that
many cells live and multiply without any visible centrosomes, that
a centrosome may disappear and be created anew, and even that
under artificial conditions abnormal chemical stimuh may lead to
the formation of new centrosomes. We may safely take it that the
centrosome, or the "attraction sphere," is essentially a "centre of
force," and that this dynamic centre may or may not be constituted
by (but will be very apt to produce) a concrete and visible con-
centration of matter.

It is far from correct to say, as is often done, that the cell- wall,
or cell-membrane, belongs "to the passive products of protoplasm
rather than to the hving cell itself"; or to say that in the animal
cell, the cell-wall, because it is "slightly developed," is relatively
unimportant compared with the important role which it assumes
in plants. On the contrary, it is quite certain that, whether visibly
diiferentiated into a semi-permeable membrane or merely con-
stituted by a liquid film, the surface of the cell is the seat of
important forces, capillary and electrical, which play an essential
part in the dynamics of the cell. Even in the thickened, largely
solidified cellulose wall of the plant-cell, apart from the mechanical
resistances which it affords, the osmotic forces developed in con-
nection with it are of essential importance.

But if the cell acts, after this fashion, as a whole, each part
interacting of necessity with the rest, the same is certainly true of

344 FORM AND STRUCTURE OF THE CELL [ch.

the entire multicellular organism : as Schwann said of old, in very-
precise and adequate words, "the whole organism subsists only by
means of the recijpTocal action of the single elementary parts*." As
Wilson says again, "the physiological autonomy of the individual
cell falls into the background . . . and the apparently composite
character which the multicellular organism may exhibit is owing to
a secondary distribution of its energies among local centres of
action!." I* is here that the homology breaks down which is so
often drawn, and overdrawn, between the unicellular organism and
the individual cell of the metazoonj.

Whitman, Adam Sedgwick §, and others have lost no opportunity
of warning us against a too hteral acceptation of the cell-theory,
against the view that the multicellular organism is a colony (or, as
Haeckel called it, in the case of the plant, a "republic") of inde-
pendent units of Ufe||. As Goethe said long ago, "Das lebendige
ist zwar in Elemente zerlegt, aber man kann es aus diesen nicht
wieder zusammenstellen und beleben " ; the dictum of the Cellular-
pathologie being just the opposite, "Jedes Thier erscheint als cine
Summe vitaler Einheiten, von denen jede den vollen Charakter des
Lebens an sich trdgt."

Hofmeister and Sachs have taught us that in the plant the growth

♦ Theory of Cells, p. 191.

t The Cell in Development, etc., p. 59; cf. 3rd ed. (1925), p. 102.-

X E.g. Brticke, Elementarorganismen, p. 387: "Wir miissen in der Zelle einen
kleinen Thierleib sehen, und diirfen die Analogien, welche zwischen ihr und den
kleinsten Thierformen existiren, niemals aus den Augen lassen."

§ C. 0. Whitman, The inadequacy of the cell-theory, Journ. Morphol. viii,
pp. 639-658, 1893; A. Sedgwick, On the inadequacy of the cellular theory of
development, Q.J. M.S. xxxvii, pp. 87-101, 1895; xxxviii, pp. 331-337, 1896.
Cf. G. C. Bourne, ibid, xxxviii, pp. 137-174, 1896; Clifford Dobell, The principles
of Protistology, Arch. f. Protistenk. xxiii, p. 270, 1911.

II Cf. 0. Hertwig, Die Zelle und die Gewebe, 1893, p. 1: "Die Zellen, in welche
der Anatom die pflanzlichen und thierischen Organismen zerlegt, sind die Trager
der Lebensfunktionen ; sie sind, wie Virchow sich ausgedruckt hat, die 'Lebensein-
heiten.' Von diesem Gesichtspunkt aus betrachtet, erscheint der Gesammtlebens-
prozess eines zusammengesetzten Organismus nichts Anderes zu sein als das hochst
yerwickelte Resultat der einzelnen Lebensprozesse seiner zahlreichen, verschieden
functionirenden Zellen." But in 1920 Doncaster (Cytology, p. 1) declared that "the
old idea of discrete and independent cells is almost abandoned," and that the
word cell was coming to be used "rather as a convenient descriptive term than
as denoting a fundamental concept of biology"; and James Gray {Experimental
Cytology, p. 2) said, in 1931, that "we must be careful to avoid any tacit assumption
that the cell is a natural, or even legitimate, unit of life and function."

IV] OF THE CELL-THEORY 345

of the mass, the growth of the organ, is the primary fact, that
"cell formation is a phenomenon very general in organic life, but
still only of secondary significance." "Comparative embryology,'*
says Whitman, "reminds us at every turn that the organism
dominates cell-formation, using for the same purpose one, several,
or many cells, massing its material and directing its movements
and shaping its organs, as if cells did not exist*." So Rauber
declared that, in the whole world of organisms, "das Ganze liefert
die Theile, nicht die Theile das Ganze: letzteres setzt die Theile
zusammen, nicht diese jenesf." And on the botanical side De Bary
has summed up the matter in an aphorism, "Die Pflanze bildet
Zellen, nicht die Zelle bildet Pflanzen."

Discussed almost wholly from the concrete, or morphological
point of view, the question has for the most part been made to turn
on whether actual protoplasmic continuity can be demonstrated
between one cell and another, whether the organism be an actual
reticulum, or syncytium J. But from the dynamical point of view
the question is much simpler. We then deal not with material
continuity, not with little bridges of connecting protoplasm, but
with a continuity of forces, a comprehensive field of force, which
runs through and* through the entire organism and is by no means
restricted in its passage to a protoplasmic continuum. And such
a continuous field of force, somehow shaping the whole organism,
independently of the number, magnitude and form of the individual
cells, which enter hke a froth into its fabric, seems to me certainly
and obviously to exist. As Whitman says, "the fact that physio-
logical unity is not broken by cell-boundaries is confirmed in so
many ways that it must be accepted as one of the fundamental
truths of biology §."

* Journ. Morph. viii, p. 653, 1893.

t Neue Grundlegungen zur Kenntniss der Zelle, Morph. Jahrb. viii, pp. 272,
313, 333, 1883.

X Cf. e.g. Ch. van Bambeke, A propos de la delimitation cellulaire, Bull. Soc.
beige de Microsc. xxiii, pp. 72-87, 1897.

§ Journ. Morph. ii, p. 49, 1889.

CHAPTER V

THE FORMS OF CELLS

Protoplasm, as we have already said, is a fluid* or a semi-fluid
substance, and we need not try to describe the particular properties
of the colloid or jelly-like 'substances to which it is alHed, or rather
the characteristics of the "colloidal state" in which it and they
exist; we should find it no easy matter f. Nor need we appeal to
precise theoretical definitions of fluidity, lest we come into a
debatable land. It is in the most general sense that protoplasm
is "fluid." As Graham said (of colloid matter in general), "its
softness partakes of fluidity, and enables the colloid to become a
vehicle for liquid diffusion, like water itself J." When we can deal
with protoplasm in sufficient quantity we see it /oi^§; particles
move freely through it, air-bubbles and liquid droplets shew round
or spherical within it; and we shall have much to say about other
phenomena manifested by its own surface, which are those especially
characteristic of liquids. It may encompass and contain solid
bodies, and it may "secrete" solid substances within or around
itself; and it often happens in the complex Hving organism that
these sohd substances, such as shell or nail or horn or feather,
remain when the protoplasm which formed them is dead and gone.
But the protoplasm itself is fluid or semi-fluid, and permits of free
(though not necessarily rapid) diffusion and easy convection of
particles within itself, which simple fact is of elementary importance

* Cf. W. Kuhne, Ueber das Protoplasma, 1864.

t Sand, or a heap of millet-seed, may in a sense be deemed a "fluid," and such
the learned Father Boscovich held them to be {Theoria, p. 427), but at best they
are fluids without a surface. Galileo had drawn the same comparison; but went on
to contrast the continuity, or infinite subdivision, of a fluid with the finite, dis-
continuous subdivision of a fine powder. Cf. Boyer, Concepts of the Calculus, 1939,
p. 291. .

J Phil Trans, clt, p 183, 1861; Researches, ed. Angus Smith, 1877, p 553.
We no longer speak, however, of "colloids' in a specific sense, as Graham did;
for any substance can be brought into the "colloidal state" by appropriate means
or in an appropriate medmm.

§ The copious protoplasm of a Myxomycete has been passed unharmed through
filter-paper with a pore-size of about 1 /x, or 0001 mm.

CH. V] OF THE COLLOID STATE 347

in connection with form, throwing light on what seem to be common
characteristics and pecuHarities of the forms of Hving things.

Much has been done, and more said, about the nature of protoplasm since
this book was written. Calling cytoplasm the cell-protoplasm after deduction
of chloroplasts and other gross inclusions, we find it to contain fats, proteins,
lecithin and some other substances combined with much water (up to
97 per cent.) to form a sort of watery gel. The microscopic structures
attributed to it, alveolar, granular or fibrillar, are inconstant or invalid;
but it does appear to possess an invisible or submicroscopic structure,
distinguishing it from an ordinary colloid gel, and forming a quasi-solid
framework or reticulum. This framework is based on proteid macromolecules,
in the form of polypeptide chains, of great length and carrying in side-chains
other organic constituents of the cytoplasm*. The polymerised units
represent the micellae f which the genius of Nageli predicted or postulated
more than sixty years ago; and we may speak of a "micellar framework"
as representing in our cytoplasm the dispersed phase of an ordinary colloid.
In short, as the cytoplasm is neither true fluid not true solid, neither is it true
colloid in the ordinary sense. Its micellar structure gives it a certain rigidity
or tendency to retain its shape, a certain plasticity and tensile strength, a
certain ductility or capacity to be drawn out in threads; but yet leaves it
with a permeability (or semi-permeability), a capacity to swell by imbibition,
above all an ability to stream and flow, which justify our calling it "fluid
or semi-fluid," and account for its exhibition of surface-tension and other
capillary phenomena.

The older naturalists, in discussing the differences between organic and
inorganic bodies, laid stress upon the circumstance that the latter grow by
"agglutination," and the former by what they termed "intussusception."
The contrast is true; but it applies rather to solid or crystalline bodies as
compared with colloids of all kinds, whether living or dead. But it so happens
that the great majority of colloids are of organic origin; and out of them our
bodies, and our food, and the very clothes we wear, are almost wholly made.

A crystal "grows" by deposition of new molecules, one by one
and layer by layer, each one superimposed on the solid substratum

* See {int. al.) A. Frey-Wyssling, Subrnikroskopische Morphologie des Protoplasmas,
Berlin, 1938; cf. Nature, June 10, 1939, p. 965; also A. R. Moore, in Scientia,
LXii, July 1, 1937. On the nature of viscid fluid threads, cf. Larmor, Nature,
July 11, 1936, p. 74.

f Micella, or micula, diminutive of mica, a crumb, grain or morsel — ynica panis,
salis, turis, etc. Nageli used the word to mean an aggregation of molecules, as
the molecule is an aggregation of atoms; the one, however, is a physical and the
other a chemical concept. Roughly speaking, we may think of micellae as varying
from about 1 to 200 /x/x; they play a corresponding part in the "disperse phase"
of a colloid to that played by the molecules in an ordinary solution. The macro-
molecules of modern chemistry are sometimes distinguished from these as still
larger aggregates. See Carl Nageli, Das Mikroskop (2nd ed.), 1877; Theorie der
Gahrung, 1879.

348 THE FORMS OF CELLS [ch.

already formed. Each particle would seem to be influenced only
by the particles in its immediate neighbourhood, and to be in
a state of freedom and independence from the influence, either
direct or indirect, of its remoter neighbours. So Lavoisier was
the first to say. And as Kelvin and others later on explained
the formation and the resulting forms of crystals, so wo beheve
that each added particle takes up its position in relation to its
immediate neighbours already arranged, in the holes and corners
that their arrangement leaves, and in closest contact with the
greatest number*; hence we may repeat or imitate this process of
arrangement, with great or apparently even with precise accuracy
(in the case of the simpler crystalhne systems), by piling up spherical
pills or grains of shot. In so doing, we must have regard to the
fact that each particle must drop into the place where it can go
most easily, or where no easier place offers. In more technical
language, each particle is free to take up, and does take up, its
position of least potential energy relative to those already there:
in other words, for each particle motion is induced until the energy
of the system is so distributed that no tendency or resultant force
remains to move it more. This has been shewn to lead to the
production of plane surfaces f (in all cases where, by the hmitation
of material, surfaces must occur); where we have planes, there
straight edges and solid angles must obviously occur also, and, if
equihbrium is to follow, must occur symmetrically. Our pihng up
of shot to make mimic crystals gives us visible demonstration that
the result is actually to obtain, as in the natural crystal, plane
surfaces and sharp angles symmetrically disposed.

* Cf. Kelvin, On the molecular tactics of a crystal, The Boyle Lecture, Oxford,
1893; Baltimore Lectures, 1904, pp. 612-642. Here Kelvin was mainly following
Bravais's (and Frankenheim's) theory of "space-lattices," but he had been largely
anticipated by the crystallographers. For an account of the development of the
subject in modern crystallography, by Sohncke, von Fedorow, 8chonfliess, Barlow
and others, see (e.g.) Tutton's Crystallography, and the many papers by W. E. Bragg
and others.

t In a homogeneous crystalline arrangement, symmetry compels a locus of one
property to be a plane or set of planes ; the locus in this case being that of least
surface potential energy. Crystals "seem to be, as it were, the Elemental Figures,
or the A B C of Nature's workmg, the reason of whose curious Geometrical Forms
(if I may so call them) is very easily explicable" (Robert Hooke, Posthumous Works^
1745, p. 280).

V] OF INTUSSUSCEPTION 349

But the living cell grows in a totally different way, very much
as a piece of glue swells up in water, by "imbibition," or by inter-
penetration into and throughout its entire substance. The semi-
fluid colloid mass takes up water, partly to combine chemically
with its individual molecules*; partly by physical diffusion into
the interstices between molecules or micellae, and partly, as it would
seem, in other ways; so that the entire phenomenon is a complex
and even an obscure onef. But, so far as we are concerned, the
net result is very simple. For the equilibrium, or tendency to
equilibrium, of fluid pressure in all parts of its interior while the
process of imbibition is going on, the constant rearrangement of its
fluid mass, the contrast in short with the crystalhne method of
growth where each particle comes to rest to move (relatively to the
whole) no more, lead the mass of jelly to swell up very much as a
bladder into which we blow air, and so, by a graded and harmonious
distribution of forces, to assume everywhere a rounded and more
or less bubble-hke external form J. So, when the same school of
older naturahsts called attention to a new distinction or contrast of
form between organic and inorganic objects, in that the contours
of the former tended to roundness and curvature, and those of the
latter to be bounded by straight lines, planes and sharp angles, we
see that this contrast was not a new and different one, but only
another aspect of their former statement, and an immediate con-
sequence of the difference between the processes of agglutination
and intussusception §.

So far then as growth goes on undisturbed by pressure or other
external force, the fluidity of the protoplasm, its mobility internal

* This is what Graham called the water of gelatination, on the analogy of water
of crystallisation; Chem. and Phys. Researches, p. 597.

t On this important phenomenon, see J. R. Katz, Oesetze der Quellung, Dresden,
1916. Swelling is due to "concentrated solution," and is accompanied by increase
of volume and liberation of energy, as when the Egyptians split granite by the
swelling of wood.

X Here, in a non- crystalline or random arrangement of particles, symmetry
ensures that the potential energy shall be the same per unit area of all surfaces;
and it follows from geometrical considerations that the total surface energy will
be least if the surface be spherical.

§ Intussusception has its shades of meaning; it is excluded from the idea of a
crystalline body, but not limited to the ordinary conception of a colloid one. When
new micellar strands become interwoven in the micro-structure of a cellulose cell-
wall, that is a special kind of "intussusception."

350 THE FORMS OF CELLS [ch.

and external*, and the way in which particles move freely hither
and thither within, all manifestly tend to the production of swelhng,
rounded surfaces, and to their great predominance over plane sur-
faces in the contours of the organism. These rounded contours
will tend to be preserved for a while, in the case of naked protoplasm
by its viscosity, and in presence of a cell-wall by its very lack of
fluidity. In a general way, the presence of curved boundary
surfaces will be especially obvious in the unicellular organisms, and
generally in the external form of all organisms, and wherever
mutual pressure between adjacent cells, or other adjacent parts,
has not come into play to flatten the rounded surfaces into planes.

The swelling of any object, organic or inorganic, living or dead, is bound to
be influenced by any lack of structural symmetry or homogeneity f. We
may take it that all elongated structures, such as hairs, fibres of silk or cotton,
fibrillae of tendon and connective tissue, have by virtue of their elongation
an invisible as well as a visible polarity. Moreover, the ultimate fibrils are
apt to be invested by a protein different from the "collagen" within, and
liable to swell more or to swell less. In ordinary tendons there is a "reticular
sheath," which swells less, and is apt to burst under pressure from within;
it breaks into short lengths, and when the strain is relieved these roll back,
and form the familiar annuli. Another instance is the tendency to swell of
the "macro- molecules" of many polymerised organic bodies, proteins among
them.

But the rounded contours which are assumed and exhibited by
a piece of hard glue when we throw it into water and see it expand
as it sucks the water up, are not near so regular nor so beautiful as
are those which appear when we blow a bubble, or form a drop, or
even pour water into an elastic bag. For these curving contours
depend upon the properties of the bag itself, of the film or membrane
which contains the mobile gas, or which contains or bounds the
mobile liquid mass. And hereby, in the case of the fluid or semifluid
mass, we are introduced to the subject of surface-tension: of which
indeed we have spoken in the preceding chapter, but which we must
now examine with greater care.

* The protoplasm of a sea-urchin's egg has a viscosity only about four times,
and that of various plants not more than ten to twenty times, that of water itself.
See, for a general discussion, L. V. Heilbrunn, Colloid Symposium Monograph^ 1928.

t D. Jordan Lloyd and R. H. Marriott, The swelling of structural proteins,
Proc. U.S. (B), No. 810, pp. 439-44.3, 1935.

V] OF SURFACE TENSION 351

Among the forces which determine the forms of cells, whether
they be sohtary or arranged in contact with one another, this force
of surface-tension is certainly of great, and is probably of paramount,
importance. But while we shall try to separate out the phenomena
which are directly due to it, we must not forget that, in each
particular case, the actual conformation which we study may be,
and usually is, the more or less complex resultant of surface-tension
acting together with gravity, mechanical pressure, osmosis, or other
physical forces. The peculiar beauty of a soap-bubble, solitary or
in collocation, depends on the absence (to all intents and purposes)
of these ahen forces from the field ; hence Plateau spoke of the films
which were the subject of his experiments as "lames fluides sans
pesanteur.'' The resulting form is in such a case so pure, and simple
that we come to look on it as wellnigh a mathematical abstraction.

Surface-tension, then, is that force by which we explain the form
of a drop or of a bubble, of the surfaces external and internal of
a "froth" or collocation of bubbles, and of many other things of
like nature and in hke circumstances*. It is a property of Hquids
(in the sense at least with which our subject is concerned), and it
is manifested at or very near the surface, where the liquid comes
into contact with another hquid, a solid or a gas. We note here
that the term surface is to be interpreted in a wide sense; for
wherever we have solid particles embedded in a fluid, wherever we
have a non-homogeneous fluid or semi-fluid, or a "two-phase colloid "
such as a particle of protoplasm, wherever we have the presence of
"impurities" as in a mass of molten metal, there we have always
to bear in mind the existence of surfaces and of surface-phenomena,
not only on the exterior of the mass but also throughout its inter-
stices, wherever like and unlike meet.

* The idea of a "surface-tension" in liquids was first enunciated by Segner, and
ascribed by him to forces of attraction whose range of action was so small "ut
nullo adhuc sensu percipi potuerat" {Defiguris super ficier um fiuidarum, in Comment.
Soc. Roy. GoUingen, 1751, p. 301). Hooke, in the Micrographia (1665, Obs. viii,
etc.), had called attention to the globular or spherical form of the little morsels
of steel struck off by a flint, and had shewn how to make a powder of such spherical
grains, by heating fine filings to melting point, "This Phaenomenon" he said
"proceeds from a propriety which belongs to all kinds of fluid Bodies more or less,
and is caused by the Incongruity of the Atnbient and included Fluid, which so
acts and modulates each other, that they acquire, as neer as is possible, a spherical
or globular form "

352 THE FORMS OF CELLS [ch.

A liquid in the mass is devoid of structure ; it is homogeneous, and
without direction or polarity. But the very concept of surface-
tension forbids this to be true of the surface-layer of a body of liquid,
or of the "interphase" between two liquids, or of any film, bubble,
drop, or capillary jet or stream. In all these cases, and more empha-
tically in the case of a "monolayer," even the liquid has a structure
of its own ; and we are reminded once again of how largely the living
organism, whether high or low, is composed of colloid matter in
precisely such forms and structural conditions.

Surface-tension is due to molecular force * : to force, that is to say,
arising from the action of one molecule upon another; and since
we can only ascribe a small "sphere of action" to each several
molecule, this force is manifested only within a narrow range.
Within the interior of the liquid mass we imagine that such molecular
interactions negative one another ; but at and near the free surface,
within a layer or film approximately equal to the range of the
molecular force — or to the radius of the aforesaid " sphere of action "
— there is a lack of equilibrium and a consequent manifestation of
force.

The action of the molecular forces has been variously explained.
But one simple explanation (or mode of statement) is that the
molecules of the surface-layer are being constantly attracted into
the interior by such as are just a little more deeply situated; the
surface shrinks as molecules keep quitting it for the interior, and
this surface-shrinkage exhibits itself as a surface-tension. The process
continues till it can go no farther, that is to say until the surface
itself becomes a "minimal areaf." This is a sufficient description
of the phenomenon in cases where a portion of liquid is subject to
no other than its own molecular forces, and (since the sphere has,

* While we explain certain phenomena of the organism by reference to atomic
or molecular forces, the following words of Du Bois Reymond's seem worth
recalling: ' Naturerkennen ist Zuriickfiihren der Veranderungen in der Korperwelt
auf Bewegung von Atomen, die durch deren von der Zeit unabhangige Centralkrafte
bewirkt werden, oder Auflosung der Naturkrafte in Mechanik der Atome. Es ist
eine psychologische Erfahrungstatsache dass, wo solche Auflosung gelangt, unser
Causalbediirfniss vorlaiifig sioh befriedigt fiihlt" (Ueber die Grenzen des Natnr-
erkennens, Leipzig, 1873).

t There must obviously be a certain kinetic energy in the molecules within
the drop, to balance the forces which are trying to contract and diminish the
surface.

V] OF SURFACE ENERGY 353

of all solids, the least surface for a given volume) it accounts for
the spherical form of the raindrop*, of the grain of shot, or of the
living cell in innumerable simple organisms f. It accounts also, as
we shall presently see, for many much more complicated forms,
manifested under less simple conditions.

Let us note in passing that surface-tension is a comparatively
small force and is easily measurable: for instance that between
water and air is equivalent to but a few grains per linear inch, or
a few grammes per metre. But this small tension, when it exists
in a curved surface of great curvature, such as that of a minute drop,
gives nse to a very great pressure, directed inwards towards the
centre of curvature. We may easily calculate this pressure, and so
satisfy ourselves that, when the radius of curvature approaches
molecular dimensions, the pressure is of the order of thousands of
atmospheres — a conclusion which is supported by other physical
considerations.

The contraction of a liquid surface, and the other phenomena of
surface-tension, involve the doing of work, and the power to do
work is what we call Energy. The whole energy of the system is
diffused throughout its molecules, as is obvious in such a simple
case as we have just considered; but of the whole stock of energy
only the part residing at or very near the surface normally manifests
itself in work, and hence we speak (though the term be open to

* Raindrops must be spherical, or they would not produce a rainbow; and the
fact that the upper part of the bow is the brightest and sharpest shews that the
higher raindrops are more truly spherical, as well as smaller than the lower ones.
So also the smallest dewdrops are found to be more iridescent than the large, shewing
that they also are the more truly spherical; cf. T. W. Backhouse, in Monthly
Meteorol. Mag. March, 1879. Mercury has a high surface-tension, and its globules
are very nearly round.

t That the offspring of a spherical cell (whether it be raindrop, plant or animal)
should be also a spherical cell, would seem to need no other explanation than that
both are of identical substance, and each subject to a similar equilibrium of
surface-forces; but the biologists have been apt to look for a subtler reason.
Giglio-Tos, speaking of a sea-urchin's dividing egg, asks why the daughter-cells
are spherical like the mother-cell, and finds the reason in "heredity": "Wenn also
die letztere (d. i. die Mutterzelle) eine spharische Form besass, so nehmen auch die
Tochterzellen dieselbe ein; ware urspriinglich eine kubische Form vorhanden,
so wurden also auch die Tochterzellen dieselbe auch aneignen. Die Ursache warum
die Tochterzellen die spharische Form anzunehmen trachten liegt darin, dasa
diese die Ur- und Grundform alter Zellen ist, sowohl bet Tieren ivie bei den Pflanzen''
[Arch.f. Entw. Mech. Li, p. 115, 1922).

354 THE FORMS OF CELLS [ch.

some objection) of a specific surface-energy. Surface-energy, and
the way it is increased and multiplied by the multiphcation of
surfaces due to the subdivision of the tissues into cells, is of the
highest interest tq the physiologist; and even the morphologist
cannot pass it by. For the one finds surface-energy present, often
perhaps paramount, in every cell of the body ; and the other may find,
if he will only look for it, the form of every solitary cell, hke that of
any other drop or bubble, related to if not controlled by capillarity.
The theory of "capillarity," or "surface-energy," has been set forth
with the utmost possible lucidity by Tait and by Clerk Maxwell, on
whom the following paragraphs are based: they having based their
teaching on that of Gauss*, who rested on Laplace.

Let E be the whole potential energy of a mass M of hquid; let
Cq be the energy per unit mass of the interior hquid (we may call it
the internal energy); and let e be the energy per unit mass for a
layer of the skin, of surface S, of thickness t, and density p (e being
what we call the surface-energy). It is obvious that the total energy
consists of the internal plus the surface-energy, and that the former
is distributed through the whole mass, minus its surface layers.
That is to say, in mathematical language,

E={M -S. l.tp) e^ + S . I^tpe.

But this is equivalent to writing :

= MeQ + S .I.tp{e- eo);

and this is as much as to say that the total energy of the system
may be taken to consist of two portions, one uniform throughout
the whole mass, and another, which is proportional on the one hand
to the amount of surface, and on the other hand to the difference
between e and Cq, that is to say to the difference between the unit
values of the internal and the surface energy.

It was Gauss who first shewed how, from the mutual attractions
between all the particles, we are led to an expression for what we

* See Gauss's Principia generalia Theoriae Figurae Fluidorum in statu equilibriif
Gottingen, 1830. The historical student will not overlook the claims to priority
of Thomas Young, in his Essay on the cohesion of fluids, Phil. Trans. 1805; see
the account given in his Life by Dean Peacock, 18.55, pp. 199-210.

V] OF SURFACE ENERGY 355

no\v' call the potential energy* of the system; and we know, as a
fundamental theorem of dynamics, as well as of molecular physics,
that the potential energy of the system tends to a minimum, and
finds in that minimum its stable equihbrium.

We see in our last equation that the term Mcq is irreducible, save
by a reduction of the mass itself. But the other term may be
diminished (1) by a reduction in the area of surface, S, or (2) by
a tendency towards equahty of e and ^q , that is to say by a diminu-
tion of the specific surface energy, e.

These then are the two methods by which the energy of the
system will manifest itself in work. The one, which is much the
more important for our purposes, leads always to a diminution of
surface, to the so-called "principle of minimal areas"; the other,
which leads to the lowering (under certain circumstances) of surface
tension, is the basis of the theory of Adsorption, to which we shall
have some occasion to refer as the modus operandi in the develop-
ment of a celi-wall, and in a variety of other histological phenomena.
In the technical phraseology of the day, the '^capacity factor" is
involved in the one case, and the "intensity factor" in the otherf.

Inasmuch as we are concerned with the form of the cell, it is the
former which becomes our main postulate: telhng us that the
energy-equations of the surface of a cell, or of the free surfaces of
cells in partial contact, or of the partition-surfaces of cells in contact
with one another, all indicate a minimum of potential energy in the
system, by which minimal condition the system is brought, ipso
facto, into equihbrium. And we shall not fail to observe, with
something more than mere historical interest and curiosity, how

* The word Energy was substituted for the old vis viva by Thomas Young early
in the nineteenth century, and was used by James Thomson, Lord Kelvin's brother,
about 1852, to mean, more generally, "capacity for doing work." The term potential,
or latent, in contrast to actual energy, in other words the distinction between " energy
of activity and energy of configuration," was proposed by Macquorn Rankine, and
suggested to him by Aristotle's use of 8vva/j.L$ and evepyeia; see Rankine's paper
On the general law of the transformation of energy, Phil. Soc. Glasgow, Jan.
5, 1853, cf. ibid. Jan. 23, 1867, and Phil. Mag. (4), xxvii, p. 404, 1864. The phrase
potential energy was at once adopted, but kinetic was substituted for actical by
Thomson and Tait.

t The capacity factor, inasmuch as it leads to diminution of surface, is responsible
for the concrescence of droplets into drops, of microcrystals into larger units, for
the flocculation of colloids, and for many other similar "changes of state."

356 THE FORMS OF CELLS [ch.

deeply and intrinsically there enter into this whole class of problems
the method of maxima and minima discovered by Fermat, the "loi
universelle de repos" of Maupertuis, "dont tous les cas d'equilibre
dans la statique ordinaire ne sont que des cas particuhers", and the
lineae curvae maximi miniynive proprietatibus gaudentes of Euler, by
which principles these old natural philosophers explained correctly
a multitude of phenomena, and drew the lines whereon the founda-
tions of great part of modern physics are well and truly laid. For
that physical laws deal with minima is very generally true, and is
highly characteristic of them. The hanging chain so hangs that the
height of its centre of gravity is a minimum ; a ray of hght takes
the path, however devious, by which the time of its journey is a
minimum ; two chemical substances in reaction so behave that their
thermodynamic potential tends to a minimum, and so on. The
natural philosophers of the eighteenth century were engrossed in
minimal problems ; and the differential equations which solve them
nowadays are among the most useful and most characteristic equa-
tions in mathematical physics.

"Voici," said Maupertuis, "dans un assez petit volume a quoi je
reduis mes ouvrages mathematiques ! " And when Lagrange, fol-
lowing Euler 's lead*, conceived the principle of least action, he
regarded it not as a metaphysical principle but as "un resultat
simple et general des lois de la mecaniquef." The principle of
least action J explains nothing, it tells us nothing of causation,
yet it illuminates a host of things. Like Maxwell's equations and
other such flashes of genius it clarifies our knowledge, adds weight
to our observations, brings order into our stock-in-trade of facts.
It embodies and extends that "law of simplicity" which Borelli
was the first to lay down: "Lex perpetua Naturae est ut agat
minimo labore, mediis et modis simplicissimis, facillimis, certis et

* Euler, Traite des Isoperimetres, Lausanne, 1744.

t Lagrange, Mecanique Analytique (2), ii, p. 188; ed. in 4to, 1788.

{ This profound conception, not less metaphysical in the outset than physical,
began in the seventeenth century with Fermat, who shewed (in 1629) that a ray
of light followed the quickest path available, or, as Leibniz put it, via omnium
facillima; it was over this principle that Voltaire quarrelled with Euler and
Maupertuis. The mathematician will think also of Hamilton's restatement of the
principle, and of its extension to the theory of probabilities by Boltzmann and
Willard Gibbs. Cf. (int. al.) A. Mayer, Geschichte des Prinzips der kleinsten Action,
1877.

vj THE MEANING OF SYMMETRY 357

tutis: evitando, quam maxime fieri potest, incommoditates et
prolixitates." The principle of least action grew up, and grew
quickly, out of cruder, narrower notions of "least time" or "least
space or distance." Nowadays it is developing into a principle of
"least action in space-time," which shall still govern and predict
the motions of the universe. The infinite perfection of Nature is
expressed and reflected in these concepts, and Aristotle's great
aphorism that "Nature does nothing in vain" lies at the bottom
of them all.

In all cases where the principle of maxima and minima comes
into play, as it conspicuously does in films at rest under surface-
tension, the configurations so produced are characterised by obvious
and remarkable symmetry'^. Such symmetry is highly characteristic
of organic forms, and is rarely absent in living things — save in such
few cases as Amoeba, where the rest and equilibrium on which
symmetry depends are likewise lacking. And if we ask what
physical equilibrium has to do with formal symmetry and structural
regularity, the reason is not far to seek, nor can it be better put
than in these words of Mach'sf: "In every symmetrical system
every deformation that tends to destroy the symmetry is com-
plemented by an equal and opposite deformation that tends to
restore it. In each deformation, positive and negative work is done.
One condition, therefore, though not an absolutely sufficient one,
that a niaximum or minimum of work corresponds to the form of
equiUbrium, is thus supj)ned by symmetry. Regularity is successive
symmetry; there is no reason, therefore, to be astonished that the
forms of equiUbrium are often symmetrical and regular."

A crystal is the perfection of symmetry and of regularity;
symmetry is displayed in its external form, and regularity revealed
in its internal lattices. Complex and obscure as the attractions,
rotations, vibrations and what not within the crystal may be, we
rest assured that the configuration, repeated again and again, of

* On the mathematical side, cf. Jacob Nteiner, Einfache Beweise der isoperi-
metrischen Hauptsatze, Abh. k. Akad. Wisa. Berlin, xxiii, pp. 116-135, 1836 (1838).
On the biological side, see {int. al.) F. M. Jaeger, Lectures on the Principle of Symmetry,
and its application to the natural sciences, Amsterdam, 1917; also F. T. Lewis,
Symmetry. . .in evolution, Amer. Nat. lvii, pp. 5-41, 1923.

f Science of Mechanics, 1902, p. 395; see also Mach's article Ueber die physika-
lische Bedeutung der Gesetze der Symmetrie, Lotos, xxi, pp. 139-147, 1871.

358 THE FORMS OF CELLS [ch.

the component atoms is precisely that for which the energy is a
minimum; and we recognise that this minimal distril^ution is of
itself tantamount to symmetry and to stability.

Moreover, the principle of least action is but a setting of a still
more universal law — that the world and all the parts thereof tend
ever to pass from less to more probable configurations; in which
the physicist recognises the principle of Clausius, or second law of
thermodynamics, and with which the biologist must somehow
reconcile the whole "theory of evolution."

As we proceed in our enquiry, and especially when we approach
the subject of tissues, or agglomerations of cells, we shall have from
time to time to call in the help of elementary mathematics. But
already, with very Httle mathematical help, we find ourselves in a
position to deal with some simple examples of organic forms.

When we melt a stick of sealing-wax in the flame, surface-tension
(which was ineffectively present in the solid but finds play in the
now fluid mass) rounds off its sharp edges into curves, so striving
tbwards a surface of minimal area ; and in like manner, by merely
melting the tip of a thin rod of glass, Hooke made the little spherical
beads which served him for a microscope*. When any drop of
protoplasm, either over all its surface or at some free end, as at the
extremity of the pseudopodium of an amoeba, is seen hkewise to
"round itself off," that is not an effect of "vital contractility," but,
as Hofmeister shewed so long ago as 1867, a simple consequence of
surface-tension; and almost immediately afterwards Engelmannf
argued on the same lines, that the forces which cause the contraction
of protoplasm in general may "be just the same as those which tend
to make every non-spherical drop of fluid become spherical." We
are not concerned here with the many theories and speculations
which would connect the phenomena of surface-tension with con-
tractility, muscular movement, or other special 'physiological func-

* Similarly, Sir David Brewster and others made powerful lenses by simply
dropping small drops of Canada balsam, castor oil, or other strongly refractive
liquids, on to a glass plate: On New Philosophical Instruments (Description of a
new fluid microscope), Edinburgh, 1813, p. 413. See also Hooke's Micrographia,
1665; and Adam's Essay on the Microscope, 1798, p. 8: "No person has carried
the use of these globules so far as Father Torre of Naples, etc." Leeuwenhoek.
on the other hand, ground his lenses with exquisite skill.

t Beitrage zur Physiologie des Protoplasma, Pfluger's Archlv, ii, p. 307, 1869.

V] OF SURFACE ACTION 359

tions, but we find ample room to trace the operation of the same
cause in producing, under conditions of rest and equiUbrium, certain
definite and inevitable forms.

It is of great importance to observe that the living cell is one
of those cases where the phenomena of surface-tension are by no
means limited to the outer surface; for within the heterogeneous
emulsion of the cell, between the protoplasm and its nuclear and
other contents, and in the "alveolar network" of the cytoplasm
itself (so far as that alveolar structure is actually present in life),
we have a multitude of interior surfaces; and, especially among
plants, we may have large internal "interfacial contacts" between
the protoplasm and its included granules, or its vacuoles filled with
the "cell-sap." Here we have a great field for surface-action; and
so long ago as 1865, Nageli and Schwendener shewed that the
streaming currents of plant cells might be plausibly explained by
this phenomenon. Even ten years earlier, Weber had remarked
upon the resemblance between the protoplasmic streamings and
the currents to be observed in certain inanimate drops for which
no cause but capillarity could be assigned*. What sort of chemical
changes lead up to, or go hand in hand with, the variations of
surface-tension in a hving cell, is a vastly important question. It
is hardly one for us to deal with ; but this at least is clear, that the
phenomenon is more complicated than its first investigators, such
as BUtschli and Quincke, ever took it to be. For the lowered
surface-tension which leads, say, to the throwing out of a pseudo-
podium, is accompanied first by local acidity, then by local
adsorption of proteins, lastly and consequently by gelation; and
this last is tantamount to the formation of "ectoplasm" — a step
in the direction of encystmentf.

The elementary case of Amoeba is none the less a complicated one.
The "amoeboid" form is the very negation of rest or of equihbrium;

* Poggendorff's Annalen, xciv, pp. 447-459, 1855. Cf. Strethill Wright, Phil.
Mag. Feb. 1860; Journ.Anat. and Physiol, i, p. 337, 1867.

t Cf. C, J. Pantin, Journ. Mar. Biol. Assoc, xiii, p. 24, 1923; Journ. Exp. Biol.
1923 and 1926; S. 0. Mast, Jo^lrn. Morph. xli, p. 347, 1926; and 0. W. Tiegs,
Surface tension and the theory of protoplasmic movement, Protoplasma, iv,
pp. 88-139, 1928. See also (int. al.) N. K. Adam, Physics and Chemistry of Surfaces,
1930; also Discussion on colloid science applied to biology (passim), Trans. Faraday
Soc. XXVI, pp. 663 seq., 1930.

360 THE F0RM8 OF CELLS [ch.

the creature is always moving, from one protean configuration to
another; its surface-tension is never constant, but continually
varies from here to there. Where the surface tension ^is greater,
that portion of the surface will contract into spherical or spheroidal
forms; where it is less, the surface will correspondingly extend.
While generally speaking the surface-energy has a minimal value,
it is not necessarily constant. It may be diminished by a rise of
temperature; it may be altered by contact with adjacent sub-
stances*, by the transport of constituent materials from the interior
to the surface, or again by actual chemical and fermentative change ;
for within the cell, the surface-energies developed about its hetero-
geneous contents will continually vary as these contents are affected
by chemical metabolism. As the colloid materials are broken down
and as the particles in suspension are diminished in size the "free
surface-energy" will be increased, but the osmotic energy will be
diminished!. Thus arise the various fluctuations of surface-tension,
and the various phenomena of amoeboid form and motion, which
Biitschli and others have reproduced or imitated by means of the
fine emulsions which constitute their "artificial amoebae."

A multitude of experiments shew how extraordinarily dehcate is
the adjustment of the surface-tension forces, and how sensitive they
are to the least change of temperature or chemical state. Thus,

* Haycraft and Carlier pointed out long ago {Proc. R.S.E. xv, pp. 220-224,
1888) that the amoeboid movements of a white blood-corpuscle are only manifested
when the corpuscle is in contact with some solid substance: while floating freely
in the plasma or serum of the blood, these corpuscles are spherical, that is to say
they are at rest and in equilibrium. The same fact was recorded anew by
Ledingham (On phagocytosis from an adsorptive point of view, Journ. Hygiene,
XII, p. 324, 1912). On the emission of pseudopodia as brought about by changes
in surface tension, see also {int. al.) J. A. Ryder, Dynamics in Evolution, 1894;
Jensen, L'eber den Geotropismus niederer Orgahismen, Pfliiger's Archiv, liii, 1893.
Jensen remarks that in Orbitolites, the pseudopodia issuing through the pores of
the shell first float freely, then as they grow longer bend over till they touch the
ground, whereupon they begin to display amoeboid and streaming motions.
\'erworn indicates {Ally. Physiol. 189o, p. 429), and Davenport says {Exper.
Morphology, ii, p. 376), that "this persistent clinging to the substratum is a
' thigmotropic ' reaction, and one which belongs clearly to the category of ' response '. "
Cf. Putter, Thigmotaxis bei Protisten, Arch. f. Physiol. 1900, Suppl. p. 247; but
it is not clear to my mind that to account for this simple phenomenon we need
invoke other factors than gravity and surface-action.

t Cf. Pauli, Allgemeine physikalische Chemie d. Zellen u. Gewebe, in Asher-Spiro's
Ergebnisse der Physiologic, 1912; Przibram, Vitalitdt, 1913, p. 6.

V] THE FORM OF AMOEBA 361

on a plate which we have warmed at one side a drop of alcohol
runs towards the warm area, a drop of oil away from it; and a
drop of water on the glass plate exhibits lively movements when
we bring into its neighbourhood a heated wire, or a glass rod dipped
in ether*. The water-colour painter makes good use of the surface-
tension effect of the minutest trace of ox-gall. When a plasmodium
of Aethalium creeps towards a damp spot or a warm spot, or
towards substances which happen to be nutritious, and creeps
away from solutions of sugar or of salt, we are dealing with pheno-
mena too often ascribed to 'purposeful' action or adaptation, but
every one of which can be paralleled by ordinary phenomena of
surface-tensiont- The soap-bubble itself is never in equilibrium:
for the simple reason that its film, Hke the protoplasm of Amoeba
or Aethalium, is exceedingly heterogeneous. Its surface-energies
vary from point to point, and chemical changes and changes of
temperature increase and magnify the variation. The surface of
the bubble is in continual movement, as more concentrated portions
of the soapy fluid make their way outwards from the deeper layers ;
it thins and it thickens, its colours change, currents are set up in
it and little bubbles glide over it; it continues in this state of
restless movement as its parts strive one with another in their
interactioAs towards unattainable equilibrium J. On reaching a
certain tenuity the bubble bursts: as is bound to happen when
the attenuated film has no longer the properties of matter in mass.

* 80 Bernstein shewed that a drop of mercury in nitric acid moves towards, or
is "attracted by," a crystal of potassium bichromate; Pfliiger's Archiv, lxxx,
p. 628, 1900. "^

t The surface-tension theory of protoplasmic movement has been denied by
many. Cf. (e.g.) H. S. Jennings, Contributions to the behaviour of the lower
organisms, Carnegie Instit. 1904, pp. 130-230; O. P. Dellinger, Locomotion of
Amoebae, etc., Journ. Exp. Zool. iii, pp. 337-3o7, 1906; also various papers by
Max Heidenhain, in Merkel u. Bonnet's Anatomische Hefte; etc.

X These motions of a liquid surface, and other still more striking movements,
such as those of a piece of camphor floating on water, were at one time ascribed
by certain physicists to a peculiar force, ^sui generis, the force epipolique of
Dutrochet; until van der Mensbrugghe shewed that differences of surface-tension
were enough to account for this whole series of phenomena (Sur la tension super-
ficielle des liquides, consideree au point de vue de certains mouvements observes
a leur surface, Mem. Cour. Acad, de Belgique, xxxiv, 1869, Phil. Mag. Sept. 1867;
cf. Plateau, Statique des Liquides, p. 283). An interesting early paper is by Dr
G. Carradini of Pisa, DelF adesione o attrazione di superficie, Mem. di Matem. e
di Fisica d. Soc. Ital. d. Sci. (Modena), xi, p. 75, xii, p. 89, 1804-5.

362 THE FORMS OF CELLS [ch.

The film becomes a mere bimolecular, or even a monomolecular,
layer; and at last we may treat it as a simple "surface of discon-
tinuity." So long as the changes due to imperfect equihbrium are
taking place very slowly, we speak of the bubble as "at rest"; it is
then, as Willard Gibbs remarks, that the characters of a film are
most striking and most sharply defined*.

So also, and surely not less than the soap-bubble, is every cell-
surface a complex affair. Face and interface have a molecular
orientation of their own, -depending both on the partition-membrane
and on the phases on either side. It is a variable orientation,
changing at short intervals of space and time; it coincides with
inconstant fields of force, electrical and other; it initiates, and
controls or catalyses, chemical reactions of great variety and
importance. In short we acknowledge and confess that, in sim-
pHfying the surface phenomena of the cell, for the time being and
for our purely morphological ends, we may be losing sight, or
making abstraction, of some of its most specific physical and
physiological characteristics.

In the case of the naked protoplasmic cell, as the amoeboid phase
is emphatically a phase of freedom and activity, of unstable equi-
librium, of chemical and physiological change, so on the other hand
does the spherical form indicate a phase of stabihty, of inactivity,
of rest. In the one phase we see unequal surface-tensions manifested
in the creeping movements of the amoeboid body, in the rounding-
off of the ends of its pseudopodia, in the flowing out of its substance
over a particle of "food," and in the current-motions in the interior
of its mass ; till, in the alternate phase, when internal homogeneity
and equilibrium have been as far as possible attained and the
potential energy of the system is at a minimum, the cell assumes a
rounded or spherical form, passes into a state of "rest," and (for a
reason which we shall presently consider) becomes at the same time
encysted f.

* On the equilibrium of heterogeneous substances, Collected Works, i, pp. 55-353;
Trans. Conn. Acad. 1876-78.

f We still speak of the naked protoplasm of Amoeba; but short, and far short,
of "encystment," there is always a certain tendency towards adsorptive action,
leading to a surface-layer, or "plasma- membrane," still semi-fluid but less fluid than
before, and different from the protoplasm within ; it was one of the first and chief
things revealed by the new technique of "micro-dissection." Little is known of

v] THE FORM OF AMOEBA 363

In their amoeboid phase the various Amoebae are just so many
varying distributions of surface-energy, and varying amounts of
surface-potential*. An ordinary floating drop is a figure of equi-
librium under conditions of which we shall soon have something to
say; and if both it and the fluid in which it floats be homogeneous
it will be a round drop, a "figure of revolution." But the least
chemical heterogeneity will cause the surface-tension to vary here
and there, and the drop to change its form accordingly. The httle
swarm-spores of many algae lose their flagella as they settle down,
and become mere drops of protoplasm for the time being; they
"put out pseudopodia" — in other words their outline changes; and
presently this amoeboid outHne grows out into characteristic lobes
or lappets, a sign of more or less symmetrical heterogeneity in the
cell-substance.

In a budding yeast-cell (Fig. 103 A), we see a definite and restricted
change of surface-tension. When a "bud" appears, whether with
or without actual growth by osmosis or otherwise
of the mass, it does so because at a certain part
of the cell-surface the tension has diminished, and
the area of that portion expands accordingly ; but
in turn the surface-tension of the expanded or ex-
truded portion makes itself felt, and the bud
rounds itself off into a more or less spherical form. ^^'

The yeast-cell with its bud is a simple example of an important
principle. Our whole treatment of cell-form in relation to surface-
tension depends on the fact (which Errera was the first to give clear
expression to) that the incipient cell-wall retains with but little
impairment the properties of a liquid filmf, and that the growing
cell, in spite of the wall by which it has begun to be surrounded,

the physical nature of this so-called membrane. It behaved more or less like a fluid
lipoid envelope, immiscible with its surroundings. It is easily injured and easily
repaired, and the well-being of the internal protoplasm is said to depend on the
maintenance of its integrity. Robert Chambers, Physical Properties of Protoplasm,
1926; The living cell as revealed by microdissection, Harvey Lectures, Ser. xxii,
1926-27; Journ. Gen. Physiol, vm, p. 369, 1926; etc.

* See (int. al.) Mary J. Hogue, The effect of media of different densities on the
shape of Amoebae, Journ. Exp. Zool. xxii, pp. 565-572, 1917. Scheel had said
in 1889 that A. radiosa is only an early stage of ^. proteus {Festschr. z. 70. Geburtstag
0. V. Kuptfer).

t Cf. infra, p. 561.

364 THE FORMS OF CELLS [ch.

behaves very much Hke a fluid drop. So, to a first approximation,
even the yeast-cell shews, by its ovoid and non-spherical form, that
it has acquired its shape under some influence other than the uniform
and symmetrical surface-tension which makes a soap-bubble into a
sphere. This oval or any other asymmetrical form, once acquired,
may be retained by virtue of the solidification and consequent
rigidity of the membrane-like wall of the cell; and, unless rigidity
ensue, it is* plain that such a conformation as that of the yeast-cell
with its attached bud could not be long retained as a figure of even
partial equilibrium. But as a matter of fact, the cell in this case
is not in equilibrium at all ; it is in process of budding, and is slowly
altering its shape by rounding off its bud. In like manner the
developing egg, through all its successive phases of form, is never
in complete equilibrium: but is constantly responding to slowly
changing conditions, by phases of partial, transitory, unstable and
conditional equihbrium.

There are innumerable solitary plant-cells, and unicellular
organisms in general, which, hke the yeast-cell, do not correspond
to any of the simple forms which may be generated under the
influence of simple and homogeneous surface-tension ; and in many
cases these forms, which we should expect to be unstable and
transitory, have become fixed and stable by reason of some com-
paratively sudden solidification of the envelope. This is the case,
for instance, in the more comphcated forms of diatoms or of desmids,
where we are dealing, in a less striking but even more curious way
than in the budding yeast-cell, not with one simple act of formation,
but with a complicated result of successive stages of localised growth,
interrupted by phases of partial consolidation. The original cell
has acquired a certain form, and then, under altering conditions
and new distributions of energy, has thickened here or weakened
there, and has grown out, or tended (as it were) tc branch, at par-
ticular points. We can often trace in each particular stage of
growth, or at each particular temporary growing point, the laws of
surface tension manifesting themselves in what is for the time being
a fluid surface ; nay more, even in the adult and completed structure
we have little difficulty in tracing and recognising (for instance in
the outline of such a desmid as Euastrum) the rounded lobes which
have successively grown or flowed out from the original rounded and

V] OF LIQUID FILMS 365

flattened cell. What we see in a many chambered foraminifer, such
as Glohigerina or Rotalia, is the same thing, save that the stages are
more separate and distinct, and the whole is carried out to greater
completeness and perfection. The little organism as a whole is not
a figure of equihbrium nor of minimal area; but each new bud or
separate chamber is such a figure, conditioned by the forces of
surface-tension, and superposed upon the complex aggregate of
similar bubbles after these latter have become consoHdated one by
one into a rigid system.

Let us now make some enquiry into the forms which a fluid
surface can assume under the mere influence of surface-tension.
In doing so we are limited to conditions under which other forces
are relatively unimportant, that is to say where the surface energy
is a considerable fraction of the whole energy of the system; and
in general this will be the case when we are dealing with portions
of liquid so small that their dimensions come within or near to what
we have called the molecular range, or, more generally, in which
the "specific surface" is large. In other words it is the small or
minute organisms, or small cellular elements of larger organisms,
whose forms will be governed by surface-tension; while the forms
of the larger organisms are due to other and non-molecular forces.
A large surface of water sets itself level because here gravity is
predominant; but the surface of water in a narrow tube is curved,
for the reason that we are here deahng with particles which he within
the range of each other's molecular forces. The like is the case with
the cell-surfaces and cell-partitions which we are about to study, and
the effect of gravity will be especially counteracted and concealed
when the object is immersed in a hquid of nearly its own density.

We have already learned, as a fundamental law of "capillarity,"
that a liquid film in equilibrium assumes a form which gives it a
minimal area under the conditions to which it is subject. These
conditions include (1) the form of the boundary, if such exist, and
(2) the pressure, if any, to which the film is subject: which pressure
is closely related to the volume of air, or of liquid, that the film
(if it be a closed one) may have to contain. In the simplest of cases,
as when we take up a soap-film on a plane wire ring, the film is
exposed to equal atmospheric pressure on both side^, and it ob-

36G THE FORMS OF CELLS [ch.

viously has its minimal area in the form of a plane. So long as our
wire ring lies in one plane (however irregular in outline), the film
stretched across it will still be in a plane; but if we bend the ring
so that it lies no longer in a plane, then our film will become curved
into a surface which may be extremely comphcated, but is still the
smallest possible surface which can be drawn continuously across
the uneven boundary.

The question of pressure involves not only external pressures
acting on the film, but also that which the film itself is capable of
exerting. For we have seen that the film is always contracting to
the utmost; and when the film is curved, this leads to a pressure
directed inwards — perpendicular, that is to say, to the surface of
the film. In the case of the soap-bubble, the uniform contraction
of whose surface has led to its spherical form, this pressure is
balanced by the pressure of the air within; and if an outlet be
given for this air, then the bubble contracts with perceptible force
until it stretches across the mouth of the tube, for instance across
the mouth of the pipe through which we have blown the bubble.
A precisely similar pressure, directed inwards, is exercised by the
surface layer of a drop of water or a globule of mercury, or by the
surface pellicle on a portion or "drop" of protoplasm. Only we
must always remember that in the soap-bubble, or the bubble which
a glass-blower blows, there is a twofold pressure as compared with
that which the surface-film exercises on the drop of liquid of which
it is a part ; for the bubble consists (unless it be so thin as to consist
of a mere layer of molecules*) of a Kquid layer, with a free surface
within and another without, and each of these two surfaces exercises
its own independent and coequal tension and its corresponding
pressure!.

If we stretch a tape upon a flat table, whatever be the tension
of the tape it obviously exercises no pressure upon the table below.
But if we stretch it over a curved surface, a cy finder for instance,
it does exercise a downward pressure; and the more curved the
surface the greater is this pressure, that is to say the greater is this
share of the entire force of tension which is resolved in the down-

* Or, more strictly speaking, unless its thickness be less than twice the range
of the molecular forces.

t It follows that the tension of a bubble, depending only on the surface-conditions,
is independent of the thickness of the film.

v] OF LIQUID FILMS 367

ward direction. In mathematical language, the pressm-e (p) varies
directly as the tension (T), and inversely as the radius of curvature
(R) : that is to say, p = T/R, per unit of surface.

If instead of a cylinder, whose curvature lies only in one direction,
we take a case of curvature in two dimensions (as for. instance a
sphere), then the effects of these two curvatures must be added
together to give the resulting pressure p: which becomes equal to
T/R + TIR\ or 1 1

R'^R'

i.= H+-*

And if in addition to the pressure p, which is due to surface-tension,
we have to take into account other pressures, p\ p", etc., due to
gravity or other forces, then we may say that the total pressure

p=/+/'+2'(l+l)

We may have to take account of the extraneous pressures in
some cases, as when we come to speak of the shape of a bird's egg ;
but in this first part of our subject we are able for the most part
to neglect them.

Our equation is an equation of equihbrium. The resistance to
compression — the pressure outwards — of our fluid mass is a constant
quantity (P); the pressure inwards, T (IjR + l/R'), is also con-
stant; and if the surface (unlike that of the mobile amoeba) be
homogeneous, so that T is everywhere equal, it follows that
1/i? + l/R' = C (a constant), throughout the whole surface in question.

Now equihbrium is reached after the surface-contraction has
done its utmost, that is to say when it has reduced the surface to
the least possible area. So we arrive at the conclusion, from the
physical side, that a surface such that IjR + IjR' = C, in other
\<^ords a surface which has the same 7nean curvature at all points,
is equivalent to a surface of minimal area for the volume enclosed f ;

* This simple but immensely important formula is due to Laplace (Mecanique
Celeste, Bk x, suppl. Theorie de Vaction capillaire, 1806).

t A surface may be "minimal" in respect of the area occupied, or of the volume
enclosed: the former being such as the surface which a soap-film forms when it
fills up a ring, whether plane or no. The geometers are apt to restrict the term
"minimal surface" to such as these, or, more generally, to all cases where the mean
curvature i» nil; the others, being only minimal with respect to the volume con-
tained, they call "surfaces of constant mean curvature."

368 THE FORMS OF CELLS [ch.

and to the same conclusion we may also come by ways purely
mathematical. The plane and the sphere are two obvious examples
of such surfaces, for in both the radius of curvature is everywhere
constant.

From the fact that we may extend a soap-film across any ring
of wire, however fantastically the wire be bent, we see that there
is no end to the number of surfaces of minimal area which may be
constructed or imagined*. While some of these are very com-
phcated indeed, others, such as a spiral or helicoid screw, are
relatively simple. But if we Umit ourselves to surfaces of revolution
(that is to say, to surfaces symmetrical about an axis), w^e find, as
Plateau was the first to shew, that those which meet the case are
few in number. They are six in all, nam-ely the plane, the sphere,
the cyhnder, the catenoid, the unduloid, and a curious surface which
Plateau called the nodoid.

A B CD

Fig. 104. Roulettes of the conic sections.

These several surfaces are all closely related, and the passage
from one to another is generally easy. Their mathematical inter-
relation is expressed by the fact (first shewn by Delaunayf, in 1841)
that the plane curves by whose revolution they are generated are
themselves generated as "roulettes" of the conic sections.

Let us imagine a straight line, or axis, on which a circle, ellipse or
other conic section rolls ; the focus of the conic section will describe
a line in some relation to the fixed axis, and this fine (or roulette),
when we rotate it around the axis, will describe in space one or
another of the six surfaces of revolution of which we are speaking.

If we imagine an elhpse so to roll on a base-hne, either of its foci
will describe a sinuous or wavy fine (Fig. 104, B) at a distance

* To fit a minimal surface to the boundary of any given closed curve in space is
a problem formulated by Lagrange, and commonly known as the "problem of
Plateau," who solved it with his soap-films.

f Sur la surface de revolution dont la courbure moyenne est constante, Journ.
de M. Liouville, vi, p. 309, 1841. Cf. {int. al.) J. Clerk Maxwell, On the theory of
rolUng curves. Trans. R.S.E. xvi, pp. 519-540, 1849; J. K. Wittemore, Minimal
surfaces of rotation, Ann. Math. (2), xix, 1917, Amer. Journ. Math, xl, p. 69,
1918; Crino Loria, Courbes planes speciales, theorie ef histoire, Milan, 574 pp., 1930.

V] OF MINIMAL SURFACES

alternately maximal and minimal from the axis; this wavy Hne,
by rotation about the axis, becomes the meridional Hne of the
surface which we call the unduloid, and the more unequal the two
axes are of our elhpse, the more pronounced will be the undulating
sinuosity of the roulette. If the two axes be equal, then our elhpse
becomes a circle ; the path described by its roUing centre is a straight
line parallel to the axis (A), and the sohd of revolution generated
therefrom will be a cylinder: in other words, the cylinder is a
"Hmiting case" of the unduloid. If one axis of our ellipse vanish,
while the other remains of finite length, then the elhpse is reduced
to a straight line with its foci at the two ends, and its roulette will
appear as a succession of semicircles touching one another upon the
axis (C); the solid of revolution will be a series of equal spheres.
If as before one axis of the ellipse vanish, but the other be infinitely
long, then the roulette described by the focus of this ellipse will be
a circular arc at an infinite distance; i.e. it will be a straight line
normal to the axis, and the surface of revolution traced by this
straight hne turning about the axis will be a plane. If we imagine
one focus of our ellipse to remain at a given distance from the axis,
but the other to become infinitely remote, that is tantamount to
saying that the elhpse becomes transformed into a parabola; and
by the rolling of this curve along the axis there is described a
catenary (D), whose solid of revolution is the catenoid.

Lastly, but this is more difficult to imagine, we have the case of
the hyperbola. We cannot well imagine the hyperbola rolling upon
a fixed straight line so that its focus shall describe a continuous
curve. But let us suppose that the fixed hne is, to begin with,
asymptotic to one branch of the hyperbola, and that the rolUng
proceeds until the hne is now asymptotic to the other branch, that
is to say touching it at an infinite distance; there will then be
mathematical continuity if we recommence rolling with this second
branch, and so in turn with the other, when each has run its course.
We shall see, on reflection, that the line traced by one and the
same focus will be an "elastic curve," describing a succession of
kinks or knots (E), and the solid of revolution described by this
meridional line about the axis is the so-called nodoid.
I

The physical transition of one of these surfaces into another can

370 THE FORMS OF CELLS [ch.

be experimentally illustrated by means of soap-bubbles, or better
still, after another method of Plateau's, by means of a large globule
of oil, supported when necessary by wire rings, and lying in a fluid
of specific gravity equal to its own.

To prepare a mixture of alcohol and water of a density precisely
equal to that of the oil-globule is a troublesome matter, and a
method devised by Mr C. R. Darhng is a great improvement on
Plateau's*. Mr Darhng used the oily hquid orthotoluidene, which
does not mix with water, has a beautiful and conspicuous red
colour, and has precisely the same density as water when both
are kept at' a temperature of 24° C. We have therefore only to
run the liquid into water at this temperature in order to produce
beautifully spherical drops of any required size : and by adding a
little salt to the lower layers of water, the drop may be made to
rest or float upon the denser liquid.

Fig. 105.

We have seen that, the soap-bubble, spherical to begin with, is
transformed into a plane when we release its internal pressure and
let the film shrink back upon the orifice of the pipe. If we blow
a bubble and then catch it up on a second pipe, so that it stretches
between, we may draw the two pipes apart, with the result that
the spheroidal surface will be gradually flattened in a longitudinal
direction, and the hubble will be transformed into a cyhnder. But
if we draw the pipes yet farther apart, the cyhnder narrows in the
middle into a sort of hour-glass form, the increasing curvature of
its transverse section being balanced by a gradually increasing
negative curvature in the longitudinal section; the cyhnder has, in
turn, been converted into an unduloid. When we hold a soft glass
tube in the flame and "draw it out," we are in the same identical
fashion converting a cylinder into an unduloid (Fig. 105, A)\ when
on the other hand we stop the end and blow, we again convert the
cylinder into an unduloid (B), but into one which is now positively,
while the former was negatively, curved. The two figures are

* See Liquid Drops and Globules, 1914, p. 11. Robert Boyle used turpentine
in much the same way; for other methods see Plateau, op. cit. p. 154.

v] OF MINIMAL SURFACES 371

essentially the same, save that the two halves of the one change
places in the other.

That spheres, cylinders and unduloids are of the commonest
occurrence among the forms of small unicellular organisms or of
individual cells in the simpler aggregates, and that in the processes
of growth, reproduction and development transitions are frequent
from one of these forms to another, is obvious to the naturalist*,
and we shall deal presently with a few of these phenomena.
But before we go further in this enquiry we must consider, to
some small extent at least, the curvatures of the six different sur-
faces, so far as to determine what modification is required, in each
case, of the general equation which apphes to them all. We shall
find that with this question is closely connected the question of
the pressures exercised by or impinging on the film, and also the
very important question of the limiting conditions which, from the
nature of the case, set bounds to the extension of certain of the
figures. The whole subject is mathematical, and we shall only deal
with it in the most elementary way.

We have seen that, in our general formula, the expression
IjR + IjR' = C, a constant; and that this is, in all cases, the
condition of our surface being one of minimal area. That is to say,
it is always true for one and all of the six surfaces which we have
to consider; but the constant C may have any value, positive,
negative or nil.

In the case of the plane, where R and R' are both infinite,
IjR + IjR' = 0. The expression therefore vanishes, and our dy-
namical equation of equihbrium becomes P = p. In short, we can
only have a plane film, or we shall only find a plane surface in our
cell, when on either side thereof we have equal pressures or no
pressure at all; a simple case is the plane^partition between two
equal and similar cells, as in a filament of Spirogyra.

In the sphere the radii are all equal, R= R'; they are also positive,
and T (l/R + l/R'), or 2T/R, is a- positive quantity, involving a
constant positive pressure P, on the other side of the equation.

In the cylinder one radius of curvature has the finite and positive
value R; but the other is infinite. Our formula becomes TjR, to

* They tend to reappear, no less obviously, in those precipitated structures which
simulate organic form in the experiments of Leduc, Herrera and Lillie.

372 THE FORMS OF CELLS [ch.

which corresponds a positive pressure P, suppHed by the surface-
tension as in the case of the sphere, but evidently of just half the
magnitude.

In plane, sphere and cyUnder the two principal curvatures are
constant, separately and together; but in the unduloid the curva-
tures change from one point to another. At the middle of one of
the swollen "beads" or bubbles, the curvatures are both positive;
the expression (l/R + 1/^') is therefore positive, and it is also finite.
The film exercises (like the cyUnder) a positive pressure inwards,
to be compensated by an equivalent outward pressure from within.
Between two adjacent beads, at the middle of one of the narrow
necks, there is obviously a much stronger curvature in the trans-
verse direction; but the total pressure is unchanged, and we now
see that a negative curvature along the unduloid balances the
increased curvature in the transverse direction. The sum of the two
must remain positive as well as constant; therefore the convex or
positive curvature must always be greater than the concave or
negative curvature at the same point, and this is plainly the case
in our figure of the unduloid.

The catenoid, in this respect a limiting case of the unduloid, has
its curvature in one direction equal and opposite to its curvature
in the other, this property holding good for all points of the surface ;
R = — R'; and the expression becomes

{l/R -+- l/R') = (L/R - l/R) = 0.

That is to say, the mean curvature is zero, and the catenoid,
like the plane itself, has no curvature, and exerts no pressure.
None of the other surfaces save these two share this remarkable
property; and it follows that we may have at times the plane and
the catenoid co-existing as parts of one and the same boundary
system, just as the cyHnder or the unduloid may be capped by
portions of spheres. It follows also that if we stretch a soap-film
between two rings, and so form an annular surface open at both ends,
that surface is a catenoid : the simplest case being when the rings are
parallel and normal to the axis of the figure*.

* A topsail bellied out by the wind is not a catenoid surface, but in vertical
section it is everywhere a catenary curve; and Diirer shews beautiful catenary
curves in the wrinkles under an Old Man's eyes. A simple experiment is to invert

v] OF FIGURES OF EQUILIBRIUM 373

The nodoid is, like the iinduloid, a continuous curve which keeps
altering its curvature as it alters its distance from the axis; but in
this case the resultant pressure inwards is negative instead of
positive. But this curve is a complicated one, and its full mathe-
matical treatment is too hard for us.

In one of Plateau's experiments, a bubble of oil (protected from
gravity by a fluid of equal density to its own) is balanced between
annuh; and by adjusting the distance apart of these, it may be
brought to assume the form of Fig. 106, that is to say, of a cyhndet
with spherical ends; there is then everywhere a pressure inwards
on the fluid contents of the bubble a pressure due to the convexity

Fig. 106.

Fig. 107.

of the surface film. This cylinder may be converted into an undu-
loid, either by drawing the rings farther apart or by abstracting
some of the oil, imtil at length rupture ensues, and the cylinder
breaks up into two spherical drops. Or again, if the surrounding
liquid be made ever so little heavier or hghter than that which
constitutes the drop, then gravity comes into play, the conditions
of equihbrium are modified accordingly, and the cylinder becomes
part of an unduloid, with its dilated portion above or below as the
case may be (Fig. 107).

In all cases the unduloid, like the original cyhnder, is capped
by spherical ends, the sign and the consequence of a positive
pressure produced by the curved walls of the unduloid. But if
our initial cyhnder, instead of being tall, be a flat or dumpy one

a small funnel in a large one, wet them with soap-solution, and draw them apart;
the film which develops between them is a catenoid surface, set perpendicularly
to the two funnels. On this and other geometrical illustrations of the fact that
a soap-film sets itself at right angles to a solid boundary, see an elegant paper by
Mary E. Sinclair, in Annals of Mathematics, viii, 1907.

374

THE FORMS OF CELLS

[CH.

(with certain definite relations of height to breadth), then new
phenomena may occur. For now, if oil be cautiously withdrawn
from the mass by help of a small syringe, the cylinder may be made
to flatten down so that its upper and lower surfaces become plane:
which is of itself a sufficient indication that the pressure inwards
is now nil. But at the very moment when the upper and lower
surfaces become plane, it will be found that the sides curve inwards,
in the fashion shewn in Fig. 108 B. This figure is a catenoid, which,

B

Fig. 108.

as we have seen, is, like the plane itself, a surface exercising no
pressure, and which therefore may coexist with the plane as part
of one and the same system.

We may continue to withdraw more oil from our bubble, drop
by drop, and now the upper and lower surfaces dimple down into
concave portions of spheres, as the
result of the negative internal
pressure ; and thereupon the peri-
pheral catenoid surface alters its
form (perhaps, on this small scale,
imperceptibly), and becomes a
portion of a nodoid. It represents,
in fact, that portion of the nodoid
which in Fig. 109 lies between such points as 0, P. While it is easy to
draw the outline, or meridional section, of the nodoid, it is obvious
that the solid of revolution to be derived from it can never be
reahsed in its entirety : for one part of the solid figure would cut, or
entangle with, another. All that we can ever do, accordingly, is to
reahse isolated portions of the nodoid*.

* This curve resembles the looped Elastic Curve (see Thomson and Tait, ii,
p. 148, fig. 7), but has its axis on the other side of the curve. The nodoid was
represented upside-down in the first edition of this book, a mistake into which others
have fallen, including no less a person than Clerk Maxwell, in his article "Capillarity "
in the Encycl. Brit. 9th ed.

Fig. 109.

V} OF FIGURES OF EQUILIBRIUM 375

In all these cases the ring or annulus is not merely a means of
mechanical restraint, controlling the form of the drop or bubble ; it
also marks the boundary, or "locus of discontinuity," between one
surface and another.

If, in a sequel to the preceding experiment of Plateau's, we use
solid discs instead of annuli, we may exert pressure on our oil-
globule as we exerted traction before. We begin again by adjusting
the pressure of these discs so that the oil assumes the form of a
cylinder: our discs, that is to say, are adjusted to exercise a
mechanical pressure just equal to what in the former case was
supplied by the surface-tension of the spherical caps or ends of the
bubble. If we now increase the pressure slightly, the peripheral
walls become convexly curved, exercising a precisely corresponding
pressure; the form assumed by the sides of our figure is now that
of a portion of an unduloid. If we increase the pressure, the
peripheral surface of oil will bulge out more and more, and will
presently constitute a portion of a sphere. But we may continue
the process yet further, and find within certain limits the system
remaining perfectly stable. What is this new curved surface which
has arisen out of the sphere, as the latter was produced from the
unduloid? It is no other than a portion of a nodoid, that part
which in Fig. 109 lies between M and N. But this surface, which is
concave in both directions towards the surface of the oil within,
is exerting a pressure upon the latter, just as did the sphere out of
which a moment ago it was transformed; and we had just stated,
in considering the previous experiment, that the pressure inwards
exerted by the nodoid was a negative one. The explanation of this
seeming discrepancy lies in the simple fact that, if we follow the
outline of our nodoid curve in Fig. 109, from OP, the surface con-
cerned in the former case, to MN, that concerned in the present,
we shall see that in the two experiments the surface of the Uquid
is not the same, but lies on the positive side of the curve in the one
case, and on the negative side in the other.

These capillary surfaces of Plateau's form a beautiful example
of the "materialisation" of mathematical law. Theory leads to
certain equations which determine the position of points in a
system, and these points we may then plot as curves on a coordinate
diagram; but a drop or a bubble may realise in an instant the

376 THE FORMS OF CELLS [ch.

whole result of our calculations, and materialise our whole ap-
paratus of curves. Such a case is what Bacon calls a "collective
instance," bearing witness to the fact that one common law is
obeyed by every point or particle of the system. Where the under-
lying equations are unknown to us, as happens in so many natural
configurations, we may still rest assured that kindred mathematical
laws are being automatically followed, and rigorously obeyed, and
sometimes half-revealed.

Of all the surfaces which we have been describing, the sphere is
the only one which can enclose space of itself; the others can only
help to do so, in combination with one another or with the sphere.
Moreover, the sphere is also, of all possible figures, that which
encloses the greatest volume with the least area of surface * ; it is
strictly and absolutely the surface of minimal area, and it is, ipso
facto, the form which will be assumed by a unicellular organism
(just as by a raindrop), if it be practically homogeneous and if, Hke
Orhulina floating in the ocean, its surroundings be likewise homo-
geneous and its field of force symmetrical f. It is only relatively
speaking that the rest of these configurations are surfaces minimae
areae', for they are so under conditions which involve various
pressures or restraints. Such restraints are imposed by the pipe or
annulus which supports and confines our oil-globule or soap-bubble ;
and in the case of the organic cell, similar restraints are supplied
by solidifications partial or complete, or other modifications local
or general, of the cell-surface or cell-wall.

One thing we must not fail to bear in mind. In the case of the
soap-bubble we look for stabihty or instability, equihbrium or non-
equilibrium, in its several configurations. But the living cell is
seldom in equihbrium. It is continually using or expending energy;
and this ceaseless flow of energy gives rise to a "steady state,"
taking the place of and simulating equilibrium. In like manner the

* On the circle and sphere as giving the smallest boundary for a given content,
see (e.g.) Jacob Steiner, Einfache Beweisen der isoperimetrischen Hauptsatze,
Berlin. Abhandlungen, 1836, pp. 123-132.

t The essential conditions of homogeneity and symmetry are none too common,
and a spherical organism is only to be looked for among simple things. The
floating (or pelagic) eggs of fishes, the spores of red seaweeds, the oospheres of
Fucus or Oedogonium, the plasma-masses escaping from the cells of Vaucheriaf
are among the instances which come to mind.

OF FIGURES OF EQUILIBRIUM

377

hardly changing outHne of a jet or waterfall is but in pseudo-
equiHbrium; it is in a steady state, dynamically speaking. Many
puzzling and apparent paradoxes of physiology, such (to take a
single instance) as the maintenance of a constant osmotic pressure
on either side of a cell-membrane, are accounted for by the fact
that energy is being spent and work done, and a steady state or
pseudo-equilibrium maintained thereby.

Before we pass to biological illustrations of our surface-tension
figures we have still another matter to deal with. We have seen
from our description of two of Plateau's classical experiments, that
at some particular point one type of surface gives place to another ;
and again we know that, when we draw out our soap-bubble into
a cyhnder, and then beyond, there comes a certain point at which
the bubble breaks in two, and leaves us with two bubbles of w^hich
each is a sphere or a portion of a sphere. In short there are certain
limits to the dimensions of our figures, within which limits equi-
librium is stable, but at which it becomes unstable, and beyond which
it breaks down. Moreover, in our composite surfaces, when the
cyhnder for instance is capped by two spherical cups or lenticular
discs, there are well-defined ratios which regulate their respective
curvatures and their respective dimensions. These two matters we
may deal with together.

Let us imagine a Hquid drop which in
appropriate conditions has been made to
assume the form of a cylinder; we have
already seen that its ends will be capped
by portions of spheres. Since one and
the same liquid film covers the sides and
ends of the drop (or since one and the
same delicate membrane encloses the
sides and ends of the cell), w^e assume
the surface-tension (T) to be everywhere
identical; and it follows, since the
internal fluid-pressure is also every-
where identical, that the expression (IjR + 1/i?') for the cylinder
is equal to the corresponding expression, which we may cafl
(1/r + 1/r'), in the case of the terminal spheres. But in the

378 THE FORMS OF CELLS [ch.

cylinder l/R' =- 0, and in the sphere 1/r = 1/r'. Therefore our
relation of equality becomes l/R = 2/r, or r = 2R; which means
that the sphere in question has just twice the radius of the cylinder
of which it forms a cap.

And if Ob, the radius of the sphere, be equal to twice the radius
(Oa) of the cyhnder, it follows that the angle aOb is an angle of 60°,
and bOc is also an angle of 60° ; that is to say, the arc be is equal to
Jtt. In other words, the spherical disc which (under the given
conditions) caps our cylinder is not a portion taken at haphazard,
but is neither more nor less than that portion of a sphere which is
subtended by a cone of 60°. Moreover, it is plain that the height
of the spherical cap, de, = Ob — ab = R (2 — ^/3) = 0'27R, where
R is the radius of our cylinder, or one-half the radius of our spherical
cap: in other words the normal height of the spherical cap over
the end of the cylindrical cell is just a very little more than one-
eighth of the diameter of the cylinder, or of the radius of the sphere.
And these are the proportions which we recognise, more or less,
under normal circumstances, in such a case as the cylindrical cell
of Spirogyra, when one end is free and capped by a portion of a
sphere*.

Among the many theoretical discoveries which we owe to Plateau,
one to which we have just referred is of peculiar importance:
namely that, with the exception of the sphere and the plane, the
surfaces with which we have been dealing are only in complete
equilibrium within certain dimensional limits, or in other words,
have a certain definite limit of stabihty; only the plane and the
sphere, or any portion of a sphere, are perfectly stable, because
they are perfectly symmetrical, figures.

Perhaps it were better to say that their symmetry is such that
any small disturbance will probably readjust itself, and leave the
plane or spherical surface as it was before, while in the other
configurations the chances are that a disturbance once set up will
travel in one direction or another, increasing as it goes. For
equihbrium and probabihty (as Boltzman told us) are nearly alHed:

* The conditions of stability of the cyhnder, and also of the catenoid, are
explained with the utmost simplicity by Clerk Maxwell, in his article, already
quoted, on "Capillarity.'' On the catenoids, see A. Terquem. C.R. xcii, pp. 407-9,
1881.

V]

OF FIGURES OF EQUILIBRIUM

379

so nearly that that state of a system which is most likely to occur,
or most likely to endure, is precisely that which we call the state of
equilibrium.

For experimental demonstration, the case of the cylinder is the
simplest. If we construct a liquid cylinder, either by drawing out
a bubble or by supporting a globule of oil between two rings, the
experiment proceeds easily until the length of the cylinder becomes
just about three times as great as its diameter. But soon afterwards
instability begins, and the cylinder alters its form; it narrows at

Fig. 111.

the waist, so passing into an unduloid, and the deformation pro-
gresses quickly until our cylinder breaks in two, and its two halves
become portions of spheres. This physical change of one surface into
another corresponds to what the mathematicians call a "discon-
tinuous solution" of a problem of minima. The theoretical limit of
stability, according to Plateau, is when the length of the cylinder is
equal to its circumference, that is to say, when L = Inr, or when the
ratio of length to diameter is represented by -n.

The fact is that any small disturbance takes the form of a wave,
and travels along the cylinder. Short waves do not affect the
stability of the system ; but waves whose length exceeds that of the
circumference tend to grow in amplitude: until, contracting here,
expanding there, the cylinder turns into a pronounced unduloid,
and soon breaks into two parts or more. Thus the cylinder is a

380 THE FORMS OF CELLS [ch.

stable figure until it becomes longer than its own circumference,
and then the risk of rupture may be said to begin. But Rayleigh
shewed that still longer waves, leading to still greater instability,
are needed to break down material resistance*. For, as Plateau
knew well, his was a theoretical result, to be departed from under
material conditions; it is affected largely by viscosity, and, as in
the case of a flowing cylinder or jet, by inertia. When inertia plays
a leading part, viscosity being small, the node of maximum in-
stability corresponds to nearly half as much again as in the simple
or theoretical case: and this result is very near to what Plateau
himself had deduced from Savart's experiments on jets of water "j".
When the fluid is external >(as when the cyhnder is of air) the wave-
length of maximal instability is longer still. Lastly, when viscosity
is very large, and becomes paramount, then the wave-length between
regions of maximal instability may become very long indeed: so
that (as Rayleigh put it) ''long (viscid) threads do not tend to divide
themselves into drops at mutual distances comparable with the
diameter of the cylinder, but rather to give way by attenuation at
few and distant places." It is this that renders possible the making
of long glass tubes, or the spinning of threads of "viscose" and like
materials; but while these latter preserve their continuity, the
principle of Plateau tends to give them something of a wavy,
unduloid surface, to the great enhancement of their beauty. We
are prepared, then, to find that such cylinders and unduloids as
occur in organic nature seldom approach in regularity to those which
theory prescribes or a soap-film may be made to shew; but rather
exhibit all manner of gradations, from something exquisitely neat
and regular to a coarse and distant approximation to the ideal
thing {.

The unduloid has certain peculiar properties as regards its limita-
tions of stabihty, but we need mention two facts only: (1) that
when the unduloid, which we produce with our soap-bubble or our

* Rayleigh, On the instability of fluid surfaces, Sci. Papers, iii, p. 594.

t Cf. E. Tylor, Phil. J^ag. xvi, pp. 504-518, 1933.

X Cf. F. Savart, Sur la constitution des veines liquides lancees par des orifices,
etc., Ann. de Chimie, liii, pp. 337-386, 1833. Rayleigh, On the instability of
a cylinder of viscous liquid, etc., Phil. Mag. (5), xxxiv, 1892, or Sci. Papers, i,
p. 361. See also Larmor, On the nature of viscid fluid threads. Nature, July 11,
1936, p. 74.

V] OF FIGURES OF EQUILIBRIUM 381

oil-globule, consists of the figure containing a complete constriction,
then it has somewhat wide limits of stabiHty; but (2) if it contain
the swollen portion, then equilibrium is limited to the case of the
figure consisting of one complete unduloid, no less nor more; that
is to say when the ends of the figure are constituted by the narrowest
portions, and its middle by the widest portion of the entire curve.
The theoretical proof of this is difficult ; but if we take the proof for
granted, the fact itself will serve to throw light on what we have
learned regarding the stabihty of the cyhnder. For, when we
remember that the meridional section of our unduloid is generated
by the rolhng of an ellipse upon a straight line in its own plane,
we easily see that the length of the entire unduloid is equal to the
circumference of the generating ellipse. As the unduloid becomes
less and less sinuous in outUne it approaches, and in time reaches,
the form of the cylinder, as a "Hmiting case"; and jpari passu, the
ellipse which generated it passes into a circle, as its foci come closer
and closer together. The cyhnder of a length equal to the circum-
ference of its generating circle is homologous to an unduloid whose
length is equal to the circumference of its generating elhpse; and
this is just what we recognise as constituting one complete segment
of the unduloid.

The cylinder turns so easily into an unduloid, and the unduloid
is capable of assuming so many graded differences of form, that we
may expect to find it abundantly and variously represented among
the simpler living things. For the same reason it is the very
stand-by of the glass-blower, whose flasks and bottles are, of
necessity, unduloids*. The blown-glass bottle is a true unduloid,
and the potter's vase a close approach to an unduloid; but the
alabaster bottle, turned on the lathe, is another story. It may be
an imitation, or a reminiscence, of the potter's or the glass-blower's
work; but it is no unduloid nor any surface of minimal area at all.

The catenoid, as we have seen, is a surface of zero pressure, and as
such is unlikely to form part (unless momentarily) of the closed
boundary of a cell. It forms a limiting case between unduloid and
nodoid, and, were it realised, it would seldom be visibly different from
the other two. In Trichodina pediculus, a minute infusorian para-

* Unless, that is to say, their shape be cramped and their mathematical beauty
annihilated, by compression in a mould.

382 THE FORMS OF CELLS [ch.

site of the freshwater polype, we have a circular disc bounded
(apparently) by two parallel rings of cilia, with a pulley-like groove

between. The groove looks very like
that catenoid surface which we have
produced from two parallel and
opposite annuli; and the fact that
the lower surface of the little creature
is practically plane, where it creeps
^.^^^^^^^^^ over the smooth body of the Hydra,
i\ '\\\ vvjN ^ looks like confirming the catenoid
Fig. 112. Triclwdiruz pediculus. analogy. But the upper surface of

the infusorian, with its ciliated
"gullet," gives no assurance of a zero pressure; and we must
take it that the equatorial groove of Trichodina resembles, or
approaches, but is not mathematically identical with, a catenoid
surface.

While those figures of equilibrium which are also surfaces of
revolution are only six in number, there is an infinite number of
other figures of equilibrium, that is to say of surfaces of constant
mean curvature, which are jiot surfaces of revolution; and it can
be shewn mathematically that any given contour can be occupied
by a finite portion of some one such surface, in stable equilibrium.
The experimental verification of this theorem lies in the simple fact
(already noted) that however we bend a wire into a closed curve,
plane or not plane, we may always fill the entire area with a con-
tinuous film. No more interesting problem has ever been pro-
pounded to mathematicians as the outcome of experiment than the
general problem so to describe a minimal surface passing through
a closed contour; and no complete solution, no general method of
approach, has yet been discovered*.

Of the regular figures of equihbrium, or surfaces of constant mean
curvature, apart from the surfaces of revolution which we have
discussed, the helicoid spiral is the most interesting to the biologist.

* Partial solutions, closely connected with recent developments of mathematical
analysis, are due to Riemann, Weierstrass and Schartz. Cf. (int. al.) G. Darboux,
Theorie des surfaces, 1914, pp. 490-601; T. Bonneson, Problemes des i^operimetres
et des iaipipJianes, Paris, 1929; Hilbert's AnschauUche Geometrie, 1932, p. 237 seq.;
a good account also in G. A. Bliss's Calculus of Variations, Chicago, 1925. See also
(int. al.) Tibor Rado, Mathem. Ztschr. xxxir, 1930; -Jesse Douglas, Amer. Math.
Journ. XXXIII, 1931, Journ. Math. Phys. xv, 1936.

V] OF FIGURES OF EQUILIBRIUM 383

This is a helicoid generated by a straight Hne perpendicular to an
axis, about which it turns at a uniform rate, while at the same time
it slides, also uniformly, along this same axis. At any point in this
surface, the curvatures are equal and of opposite sign, and the sum
of the curvatures is accordingly nil. Among what are called "ruled
surfaces," or surfaces capable of being defined by a system of
stretched strings*, the plane and the hehcoid are the only two whose
mean curvature is null, while the cyhnder is the only one whose
curvature is finite and constant. As this simplest of hehcoids
corresponds, in three dimensions, to what in two dimensions is
merely a plane (the latter being generated by the rotation of a
straight line about an axis without the superadded gliding motion
which generates the hehcoid), so there are other and much more
complicated helicoids which correspond to the sphere, the unduloid
and the rest of our figures of revolution, the generating planes of
these latter being supposed to wind spirally about an axis. In the
case of the cyhnder it is obvious that the resulting figure is indis-
tinguishable from the cyhnder itself. In the case of the unduloid
we obtain a grooved spiral, and we meet with something very like
it in nature (for instance in Spirochaetes, Bodo gracilis, etc.) ; but in
point of fact, the screw motion given to an unduloid or catenary
curve fails to give a minimal screw surface, as we might have
expected it to do.

The foregoing considerations deal with a small part only of the
theory of surface-tension, or capillarity: with that part, namely,
which relates to the surfaces capable of subsisting in equilibrium
under the action of that force, either of itself or subject to certain
simple constraints. And as yet we have limited ourselves to the
case of a single surface, or of a single drop or bubble, leaving to
another occasion a discussion of the forms assumed when such drops
or vesicles meet and combine together. Ip short, what we have
said may help us to understand the form of a cell — considered, as
with certain hmitations we may legitimately consider it, as a Hquid
drop or liquid vesicle ; the conformation of a tissue or cell-aggregate
must be dealt with in the light of another series of theoretical con-
siderations. In both cases, we can do no more than touch on the
fringe of a large and difficult subject. There are many forms

* Or rather, surfaces such that through every point there runs a straight line
which lies wholly in the surface.

384 THE FORMS OF CELLS [ch.

capable of realisation under surface-tension, and many of them
doubtless to be recognised among organisms, which we cannot
deal with in this elementary account. The subject is a very
general one; it is, in its essence, more mathematical than physical;
it is part of the mathematics of surfaces, and only comes into relation
with surface-tension because this physical phenomenon illustrates
and exemplifies, in a concrete way, the simple an4 symmetrical
conditions with which the mathematical theory is capable of deahng.
And before we pass to illustrate the physical phenomena by biological
examples, we must repeat that the simple physical conditions which
we presuppose will never be wholly realised in the organic cell.
Its substance will never be a perfect fluid, and hence equilibrium
will be slowly reached; its surface will seldom be perfectly homo-
geneous, and therefore equilibrium will seldom be perfectly attained ;
it will very often, or generally, be the seat of other forces, symmetrical
or unsymmetrical ; and all these causes will more or less perturb the
surface-tension effects*. But we shall find that, on the whole, these
effects of surface-tension though modified are not obliterated nor
even masked; and accordingly the phenomena to which I have
devoted the foregoing pages will be found manifestly recurring and
repeating themselves among the phenomena of the organic cell.

In a spider's web we find exemplified several of the principles of
surface-tension which we have now explained. The thread is spun
out of a glandular secretion which issues from the spider's body as
a semi-fluid cyHnder, the force of expulsion giving it its length and
that' of surface-tension giving it its circular section. It is too viscid,
and too soon hardened on exposure to the air, to break up into drops
or spherules ; but it is otherwise with another sticky secretion which,
coming from another gland, is simultaneously poured over the

* That "every particular that worketh any effect is a thing compounded more
or less of diverse single natures, more manifest and more obscure" is a point made
and dwelt on by Bacon. Of the same principle a great astronomer speaks as
follows: "It is one of the fundamental characteristics of natural science that we
never get beyond an approximation. . .Nature never offers us simple and undivided
phenomena to observe, but always infinitely complex compounds of many different
phenomena. Each single phenomenon can be described mathematically in terms
of the accepted fundamental laws of Nature : . . . but we can never be sure that we
have carried the analysis to its full exhaustion, and have isolated one single simple
phenomenon." W. de Sitter, in Nature, Jan. 21, 1928, p. 99.

V] OF SPIDEKS' WEBS 385

slacker cross-threads as they issue to form the spiral portion of the
web. This latter secretion is more fluid than the first, and only
dries up after several hours*. By capillarity it "wets" the thread,
spreading over it in an even film or liquid cylinder. As such it
has its limits of stabihty, and tends to disrupt at points more
distant than the theoretical wave-length, owing to the imperfect
fluidity of the viscous film and still more to the frictional drag of
the inner thread with which it is in contact. Save for this qualifi-
cation the cyhnder disrupts in the usual manner, passing first into
the wavy outhne of an unduloid, whose swollen internodes swell
more and more till the necks between them break asunder, and leave
a row of spherical drops or beads strung like dewdrops at regular
intervals along the thread. If we try to varnish a thin taut wire
we produce automatically the same identical result t; imless our
varnish be such as to dry almost instantaneously it gathers into
beads, and do what we will we fail to spread it smooth. It follows
that, according to the drying quahties of our varnish, the process
may stop at any point short of the formation of perfect spherules;
and as our final stage we may only obtain half-formed beads or the
wavy outlines of an unduloid. The beads may be helped to form
by jerking the stretched thread, and so disturbing the unstable
equilibrium of the viscid cyhnder. This the spider has been said
to do, but Dr G. T. Bennett assures me that she does nothing of the
kind. She only draws her thread out a little, and leaves it a trifle
slack ; if the gum should break into droplets, well, and good, but it
matters httle. The web with its sticky threads is not improved
thereby. Another curious phenomenon here presents itself.

In Plateau's experimental separation of a cyhnder of oil into two
spherical halves, it was noticed that, when contact was nearly
broken, that is to say when the narrow neck of the unduloid had
become very thin, the two spherical bullae, instead of absorbing
the fluid out of the narrow neck into themselves as they had done
with the preceding portion, drew out this small remaining part of

* When we see a web bespangled with dew of a morning, the dewdrops are not
drops of pure water, but of water mixed with the sticky, gummy fluid of the cross-
threads ; the radii seldom if ever shew dewdrops. See F, Strehlke, Beobachtungen
an Spinnengewebe, Poggendorff's Annalen, XL, p. 146, 1937.

t Felix Plateau recommends the use of a weighted thread or plumb-line, to be
drawn up slowly out of a jar of water or oil; Phil. Mag. xxxiv, p. 246, 1867.

386 THE FORMS OF CELLS [ch.

the liquid into a thin thread as they completed their spherical f^rm
and receded from one another: the reason being that, after the
thread or "neck" has reached a certain tenuity, internal friction
prevents or retards a rapid exit of the fluid from the thread to the
adjacent spherule. It is for the same reason that we are able to
draw a glass rod or tube, which we have heated in the middle, into
a long and uniform cyhnder or thread by quickly separating the
two ends. But in the case of the glass rod the long thin thread
quickly cools and sohdifies, while in the ordinary separation of a
liquid cylinder the corresponding intermediate cylinder remains
liquid; and therefore, Hke any other liquid cylinder, it is liable to

Fig. 113. Dew-drops on a spider's web.

break up, provided that its dimensions exceed the limit of stability.
And its length is generally such that it breaks at two points, thus
leaving two terminal portions continuous and confluent with the
spheres, and one median portion w^hich resolves itself into .a
tiny spherical drop, midway between the original and larger two.
Occasionally, the same process of formation of a connecting thread
repeats itself a second time, between the small intermediate spherule
and the large spheres; and in this case we obtain two additional
spherules, still smaller in size, and lying one on either side of our
first little one. This whole phenomenon, of equal and regularly
interspaced beads, often with little beads regularly interspaced
between the larger ones, and now and then with a third order of
still smaller beads regularly intercalated, may be easily observed
in a spider's web, such as that of Epeira, very often with beautiful
regularity — sometimes interrupted and disturbed by a sKght want
of homogeneity in the secreted fluid ; and the same phenomenon is

V] OF BEADS OR GLOBULES 387

repeated on a grosser scale when the web is bespangled with dew,
and its threads bestrung with pearls innumerable. To the older
naturalists, these regularly arranged and beautifully formed globules
on the spider's -web were a frequent source of wonderment. Black-
wall, counting some twenty globules in a tenth of an inch, calculated
that a large garden-spider's web should comprise about 120,000
globules; the net was spun and finished in about forty minutes,
and Blackwall was filled with admiration of the skill and quickness
with which the spider manufactured these little beads. And no
wonder, for according to the above estimate they had to be made
at the rate of about 50 per second*.

Here we see exemphfied what Plateau told us of the law of minimal
areas transforming the cylinder into the unduloid and disrupting it

Fig. 114. Root-hair of Trianea, in glycerine. After Berthold.

into spheres. The httle dehcate beads which stud, the long thin
pseudopodia.o.f a foraminifer, such as Gromia, or which appear in
like manner on the film of protoplasm coating the long radiating
spicules of Globigerina, represent an identical phenomenon. Indeed
we may study in a protoplasmic filament the whole process of
formation of such beads: if we squeeze out on a shde the viscid
contents of a mistletoe-berry, the long sticky threads into which the
substance runs shew the whole phenomenon particularly well. True,
many long cylindrical cells, such as are common in plants, shew no
sign of beading or disruption ; but here .the cell- walls are never fluid
but harden as they grow, and the protoplasm within is kept in place
and shape by its contact with the cell-wall. It was noticed many
years ago by Hofmeisterf, and afterwards explained by Berthold,
that if we dip the long root-hairs of certain water-plants, such as
Hydrocharis or Trianea, in a denser fluid (a httle sugar-solution or

* J. Blackwall, Spiders of Great Britain (Ray Society), 1859, p. 10; Trans. Linn.
Soc. XVI, p. 477, 1833. On the strength and elasticity of the spider's web, see
J. R. Benton, Amer. Journ. Science, xxiv, pp. 75-78, 1907.

t Lehrbuch von der Pflanzenzelley p. 71 ; cf. Nageli, Pflanzenphysiologische Unter-
suchungen [Spirogyra), ni, p. 10.

388 THE FORMS OF CELLS [ch.

dilute glycerine), the cell-sap tends to diffuse outwards, the proto-
plasm parts company with its surrounding and supporting wall, and
then hes free as a protoplasmic cylinder in the interior of the cell.
Thereupon it soon shews signs of instability, and commences to
disrupt; it tends to gather into spheres, which however, as in our
illustration, may be prevented by their narrow quarters from
assuming the complete spherical form; and in between these
spheres, we have more or less regularly alternate ones, of smaller
size*. We could not wish for a better or a simpler proof of the
essential fluidity of the protoplasm f. Similar, but less regular,
beads or droplets may be caused to appear, under stimulation by an
alternating current, in. the protoplasmic threads within the living
cells of the hairs of Tradescantia; the explanation usually given is,
that the viscosity of the protoplasm is reduced, or its fluidity
increased ; but an increase of the surface-tension would seem a more
likely reason}.

In one of Robert Chambers's delicate experiments, a filament of
protoplasm is drawn off, by a micro-needle, from the fluid surface
of a starfish-egg. If drawn too far it breaks, and part returns within
the protoplasm while the other rounds itself off on the needle's
point. If drawn out less far, it looks hke a row of beads or chain
of droplets; if yet more relaxed, the droplets begin to fuse until
the whole filament is withdrawn; if drawn out anew the process
repeats itself. The whole story is a perfect description of the
behaviour of a fltiid jet or cy finder, of varying length and
thickness §.

We may take note here of a remarkable series of phenomena,
which, though they seem at first sight to be of a very different order,

* The intermediate spherules appear with great regularity and beauty whenever
a liquid jet breaks up into drops. So a bursting soap-bubble scatters a shower
of droplets all around, sometimes all alike, but often with a beautiful alternation
of great and small. How the breaking up of thread or jet into drops may be helped,
regularised, and sometimes complicated, by external vibrations is another and by
no means unimportant story.

t Though doubtless to speak of the viscid thread as a fluid is but a first approxi-
mation; cf. Larmor, in Nature, July 11, 1936.

X Kiihne, Untersuchungen ilber das Protoplasma, 1864, p. 75, etc.

§ Cf. R. Chambers in Colloid Chemistry, theoretical and applied, ii, cap. 24, 1928;
also Ann. de Physiol, vi, p. 234, 1930; etc.

V] THE SHAPE OF A SPLASH 389

are closely related to those which attend and which bring about the
breaking-up of a liquid cylinder or thread.

In Mr Worthington's beautiful experiments on splashes*, it was
found that the fall of a found pebble into water from a height first
formed a dip or hollow in the surface, and then caused a filmy
"cup" of water to rise up all round, opening out trumpet-fashion

Fig. 115. Phases of a splash. From Worthington.

""''^WiW^JSSM,

''S^^s?*^

Fig. 116. A wave breaking into spray.

or closing in like a bubble, according to the height from which the
pebble fell. The cup or "crater" tends to be fluted in alternate
ridges and grooves, its edges get scolloped into corresponding lobes
and notches, and the projecting lobes or prominences tend to break
off or break up into drops or beads (Fig. 115). A similar appearance
is seen on a great scale in the edge of a breaking wave : for the smooth

* A Study of SplasJies, 1908, p. 38, etc.; also various papers in Proc. R.S.
1876-1882, and Phil Trans. (A), 1897 and 1900.

390 THE FORMS OF CELLS [ch.

edge becomes notched or sinuous, and the surface near by becomes
ribbed or fluted, owing to the internal flow being helped here and
hindered there by a viscous shear; and then all of a sudden the
uneven edge shoots out an array of tiny jets, which break up into
the countless droplets which constitute "spray" (Fig. 116). The
naturalist may be reminded also of the beautifully symmetrical
notching of the calycles of many hydroid zoophytes, which little
cups had begun their existence as Hquid or semi-hquid films before
they became stiff and rigid. The next phase of the splash (with
which we are less directly concerned) is that the crater subsides,
and where it stood a tall column rises up, which also tends, if it be
tall enough, to break up into drops. Lastly the column sinks down
in its turn, and a ripple runs out from where it stood.

The edge of our little cup forms a hquid ring or annulus, com-
parable on the one hand to the edge of an advancing wave, and
on the other to a hquid thread or cylinder if only we conceive the
thread to be bent round into a ring; and accordingly, just as the
thread segments first into an unduloid and then into separate
spherical drops, so likewise will the edge of cup or annulus tend to
do. This phase of notching, or beading, of the edge of the splash
is beautifully seen in many of Worthington's experiments*, and still
more beautifully in recent work (Frontispiece f). In the second place
the fact that the crater rises up means that liquid is flowing in from
below; the segmentation of the rim means that channels of easier
flow are being created, along which the liquid is led or driven into
the protuberances; and these last are thereby exaggerated into the
jets or streams which become conspicuous at the edge of the crater.
In short any film or film-like fluid or semi-fluid cup will be unstable ;
its instability will tend to show itself in a fluting of the surface and
a notching of the edge; and just such a fluting and notching are
conspicuous features of many minute organic cup-like structures.
In the hydroids (Fig. 117), we see that these common features of the

* Cf. A Study of Splashes, pp. 17, 77. The same phenomenon is often well seen
in the splash of an oar. It is beautifully and continuously evident when a strong
jet of water from a tap impinges on a curved surface and then shoots off again.

t We owe this picture to the kindness of Mr Harold E. Edgerton, of the
Massachusetts Institute of Technology. It shews the splash caused by a drop
falling into a thin layer of milk; a second drop of milk is seen above, following
the first. The exposure-time was 1/50,000 of a second.

The latter phase of a splash: the crater has subsided, a columnar jet has risen up,

and the jet is dividing into droplets. From Harold E. Edgerton,

Massachusetts Institute of Technology

V]

THE SHAPE O'F A SPLASH

391

cup and the annulation of the stem are phenomena of the same order.
A cord-hke thickening of the edge of the cup is a variant of the
same order of phenomena; it is due to the checking at the rim of
the flow of hquid from below, and a similar thickening is to be seen,
not only in some hydroid calycles but also in many Vorticellae
(cf. Fig. 124) and other cup-shaped organisms. And these are by
no means the only manifestations of surface-tension in a splash
which shew resemblances and analogies to organic form*.

The phenomena of an ordinary liquid splash are so swiftly tran-
sitory that their study is only rendered possible by photography:

01 [VA~/"'^^^VY

Fig. 117. Calycles of Campanularia spp.

but this excessive rapidity is not an essential part of the pheno-
menon. For instance, we can repeat and demonstrate many of the
simpler phenomena, in a permanent or quasi-permanent form, by
splashing water on to a surface of dry sandf, or by firing a bullet
into a soft metal target. There is nothing, then, to prevent a slow
and lasting manifestation, in a viscous medium such as a proto-
plasmic organism, of phenomena which appear and disappear with

* The. same phenomena are modified in various ways, and the drops are given
off much more freely, when the splash takes place in an electric field — all owing
to the general instability of an electrified liquid -surface; and a study of this aspect
of the subject might suggest yet more analogies with organic form. Cf. J. Zeleny,
Phys. Rev. x, 1917; J. P. Gott, Proc. Cambridge Philos. Soc. xxxi, 1935; etc.

t We find now and then in certain brick-clays of glacial origin, hard, quoit-
shaped rings, each with an equally indurated, round or flattened ball resting on it.
These may be precisely imitated by splashing large drops of water on a smooth
surface of fine dry sand. The ring corresponds, apparently, to the crater of the
splash, and the ball (or its water content) to the pillar rising in the middle.

392 THE FORMS OF CELLS [ch.

evanescent rapidity in a more mobile liquid. Nor is there anything
pecuhar in the splash itself; it is simply a convenient method of
setting up certain motions or currents, and producing certain surface-
forms, in a Hquid medium — or even in such an imperfect fluid as a bed
of sand. Accordingly, we have a large range of possible conditions
under which the organism might conceivably display configurations
analogous to, or identical with, those which Mr Worthington has
shewn us how to exhibit by one particular experimental method.

To one who has watched the potter at his wheel, it is plain that
the potter's thumb, like the glass-blower's blast of air, depends for
its efficacy upon the physical properties of the clay or "shp" it
works on, which for the time being is essentially a fluid. The cup
and the saucer, like the tube and the bulb, display (in their simple
and primitive forms) beautiful surfaces of equilibrium as manifested
under certain hmiting conditions. They are neither more nor less
than glorified "splashes," formed slowly, under conditions of
restraint which enhance or reveal their mathematical symmetry.
We have seen, and we shall see again before we are done, that the
art of the glass-blower is full of lessons for the naturahst as also
for the physicist: illustrating as it does the development of a host
of mathematical configurations and organic conformations which
depend essentially on the establishment of a constant and uniform
pressure within a closed elastic shell or fluid envelope or bubble.
In hke manner the potter's art illustrates the somewhat obscurer
and more complex problems (scarcely less frequent in biology) of a
figure of equilibrium which is an open surface of revolution. The
two series of problems are closely akin; for the glass-blower can
make most things which the potter makes, by cutting off portions
of his hollow ware; besides, when this fails and the glass-blower,
ceasing to blow, begins to use his rod to trim the sides or turn the
edges of wineglass or of beaker, he is merely borrowing a trick from
the still older craft of the potter.

It would seem venturesome to extend our comparison with these
liquid surface-tension phenomena from the cup or calycleof the
hydrozoon to the little hydroid polyp within: and yet there is
something to be learned by such a comparison. The cylindrical
body of the tiny polyp, the jet-hke row of tentacles, the beaded

V] THE SPLASH AND THE BUBBLE 393

annulations which these tentacles exhibit, the web-like film which
sometimes (when they stand a httle way apart) conjoins their bases,
the thin annular film of tissue which surrounds the little organism's
mouth, and the manner in which this annular "peristome" con-
tracts*, like a shrinking soap-bubble, to close the aperture, are
every one of them features to which we may find a singular and
striking parallel in the surface-tension phenomena of the splash "f".

Some seventy years ago much interest was aroused by Helmholtz's
work (and also Kirchhoff's) on "discontinuous motions of a fluidj";
that is to say, on the movements of one body of fluid within another,
and the resulting phenomena due to friction at the surfaces between.
What Kelvin§ called Helmholtz's "admirable discovery of the law
of vortex-motion in 'a perfect fluid" was the chief result of this
investigation; and was followed by much experimental work, in
order to illustrate and to extend the mathematical conclusions.

The drop, the bubble and the splash are parts of a long story;
and a "falling drop," or a drop moving through surrounding fluid,
is a case deserving to be considered. A drop of water, tinged with
fuchsin, is gently released (under a pressure of a couple of milh-
metres) at the bottom of a glass of water || . Its momentum enables
it to rise through a few centimetres of the surrounding water, and
in doing so it communicates motion to the water around. In front
the rising drop thrusts its way through, almost like a sohd body;
behind it tends to drag the surrounding water after it, by fluid
friction^ ; and these two motions together give rise to beautiful vorti-
coid configurations, the Strdmungspilze or Tintenpilze of their first
discoverers (Fig. 119). Under a higher and more continuous pressure

* See a Study of Splashes, p. 54.

t There is little or no difference between a splash and a burst bubble. The craters
of the moon have been compared with, and explained by, both of these.

X Helmholtz, in Berlin. Monatsber. 1868, pp. 215-228; Kirchhoflf, in CreUe'a
Journal, lxx, pp. 289-298, lxxi, 237-273, 1869-70.

§ W. Thomson, in Proc. R.S.E. vi, p. 94, 1867.

II See A. Overbeck, Ucber discontinuirliche Fliissigkeitsbewegungen, Wiedemann'' s
Annalen, ii, 1877; W. Bezold, Ueber Stromungsfiguren in Fliissigkeiten, ibid.
XXIV, pp. 569-593, 1885; P. Czermak, ibid, l, p. 329, 1893; etc.

% The frictional drag on the hinder part of the drop is felt alike in the ship, the
bird and the aeroplane, and tends to produce retarding vortices in them all. It is
always minimised in one way or another, and it is autoraaticaUy minimised in the
present instance, as the drop thins off and tapers down.

394

THE FORMS OF CELLS

[CH.

__Md3:::,,^^

Fig. 118. a, b. More phases of a splash, after Worthington.
c. A hydroid polype, after Allman.

o

^t

Fip. 119. -Liiquid jets. From A. Overbeck.

V] OF FALLING DROPS 395

the drop becomes a jet; the, form of the vortex is modified thereby,
and may be further modified by sHght differences of temperature
(i.e. of density), or by interrupting the rate of flow. To let a drop
of ink fall into water is a simple and most beautiful experiment*.
The effect is more violent than in the former case. The descending

Fig. 120. Falling drops. A, ink in water, after J. J. Thomson and Newall.
B, fusel oil in paraffin, after Tomlinson.

drop turns into a complete vortex-ring ; it expands and attenuates ;
it waves about, and the descending loops again turn into incipient
vortices (Fig. 120).

Lastly, instead of letting our drop rise or fall freely, we may use
a hanging drop, which, while it sinks, remains suspended to the
surface. Thus it cannot form a complete annulus, but only a

* J. J. Thomson and H. F. Newall, On the formation of vortex-rings by drops,
Proc. R.S. XXXIX, pp. 417-436, 1885. Emil Hatschek, On forms assumed by a
gelatinising liquid in various coagulating solutions, ibid. (A) xciv, pp. 303-316, 1918.

396 THE FORMS OF CELLS [ch.

partial vortex suspended by a thread or column — just as in Over-
beck's jet-experiments; and the figure so produced, in either case,
is closely analogous to that of a medusa or jellyfish, with its bell
or "umbrella," and its clapper or "manubrium" as well. Some
years ago Emil Hatschek made such vortex-drops as these of hquid
gelatine dropped into a hardening fluid. These " artificial medusae "
sometimes show a symmetrical pattern of radial "ribs", due to
shrinkage, and this to dehydration by the coagulating fluid. An

Fig. 121. Various medusoids: 1, Syncoryne; 2, Cordylophora;
3, Cladonema (after Allmaii).

extremely curious result of Hatschek's experiments is to shew how
sensitive these vorticoid drops are to physical conditions. For using
the same gelatine all the while, and merely varying the density of
the fluid in the third decimal place, we obtain a whole range of
configurations, from the ordinary hanging drop to the same with a
ribbed pattern, and then to medusoid vortices of various graded forms.
The living medusa has a geometrical symmetry so marked and regular
as to suggest a physical or mechanical element in the little creature's
growth and construction. It has, to begin with, its vortex-hke bell
or umbrella, with its cylindrical handle or manubrium. The bell is

OF VORTICOID OR MEDUSOID DROPS

397

traversed by radial canals, four or in multiples of four; its edge is
beset with tentacles, smooth or often beaded, at regular intervals
and of graded sizes ; and certain sensory structures, including sohd
concretions or "otohths," are also symmetrically interspaced. No
sooner made, than it begins to pulsate ; the little bell begins to " ring."

m

&

Fig. 1216. "Medusoid drops", of gelatin. After Hatschek,

Buds, miniature replicas of the parent-organism, are very apt to appear
on the tentacles, or on the manubrium or sometimes on the edge of the
bell; we seem to see one vortex producing others before our eyes.
The development of a medusoid deserves to be studied without
prejudice, from this point of view. Certain it is that the tiny
medusoids of Obelia, for instance, are , budded off with a rapidity
and a complete perfection which suggest an automatic and all but
instantaneous act of conformation, rather than a gradual process of
growth.

Moreover, not only do we recognise in a vorti-
coid drop a "schema" or analogue of medusoid
form, but we seem able to discover -various actual
phases of the splash ot drop in the all but in-
numerable living types of jellyfish; in Cladoneyna
we seem to see an early stage of a breaking drop,
and in Cordylophom a beautiful picture of incipient
vortices. It is hard indeed to say how much or little all these
analogies imply. But they indicate, at the very least, how certain
simple organic forms might be naturally assumed by one fluid mass
within another, when gravity, surface tension and fluid friction play

Fig. 122. Meckbsach-
loris, a ciliate
infusoiia.

398 THE FORMS OF CELLS [ch.

their part, under balanced conditions of temperature, density and
chemical composition.

A little green infusorian from the Baltic Sea is, as near as may
be, a medusa in miniature*. It is curious indeed to find the same
medusoid, or as we may now call it vorticoid, configuration occurring
in a form so much lower in the scale, and so much less in order of
magnitude, than the ordinary medusae.

According to Plateau, the viscidity of the liquid, while it
retards the breaking up of the cylinder and increases the length
of the segments beyond that which theory demands, has never-
theless less influence in this direction than we might have expected.
On the other hand any external support or adhesion, or mere
contact with a sohd body, will be equivalent to a reduction of
surface-tension and so will very greatly increase the stability of
our cylinder. It is for this reason that the mercury in our thermo-
meters seldom separates into drops: though it sometimes does so,
much to our inconvenience. And again it is for this reason that
the protoplasm in a long tubular or cyhndrical cell need not divide
into separate cells and internodes until the length of these far
exceeds the theoretical limits.

An interesting case is that of a viscous drop immersed in another
viscous fluid, and drawn out into a thread by a shearing motion of
the latter. The thread seems stable at first, but when left to rest
it breaks up into drops of a very definite and uniform, size, the size
of the drops, or wave-length of the unduloid of which they are made,
depending on the relative viscosities of the two threads f-

Plateau's results, though discovered by way of experiment and
though (as we have said) they illustrate the " materiahsation " of
mathematical law, are nevertheless essentially theoretical results
approached rather than realised in material systems. That a hquid
cylinder begins to be unstable when its length exceeds ^-nr is all
but mathematically true of an all but immaterial soap-bubble ; but
very far from true, as Plateau himself was well aware, in a flowing
jet, retarded by viscosity and by inertia. The principle is true and
universal; but our living cylinders do not follow the abstract laws

* Medusachloris phiale, of A. Pascher, Biol. Centralbl. xxxvii, pp. 421-429, 1917.

t See especially Rayleigh, Phil. Mag. xxxiv, p. 145, 1892, by whom the subject
is carried much further than where Plateau left it. See also {int. al.) G. I. Taylor,
Proc. R.S. (A), cxLvi, p. 501, 1934; S. Tomotika, ibid, cl, p. 322, 1935; etc.

V] OF VISCOUS THREADS 399

of mathematics, any more than do the drops and jets of ordinary
fluids or the quickly drawn and quickly cooling tubes in the glass-
worker's hands.

Plateau says that in most hquids the influence of viscosity is such
as to cause the cylinder to segment when its length is about four
times, or even six times, its diameter, instead of a fraction over
three times, as theory would demand of a perfect fluid. If we take
it at four times, the resulting spherules would have a diameter of
about 1-8 times, and their distance apart would be about 2-2 times,
the original diameter of the cylinder; and the calculation is not
difficult which would shew how these dimensions are altered in the
case of a cylinder formed around a solid core, as in the case of a
spider's web. Plateau also observed that the time taken in the
division of the cyUnder is directly proportional to its diameter,
while varying with the nature of the hquid. This question, of the
time taken in the division of a cell or filament in relation to its
dimensions, has not so far as I know been enquired into by biologists.

From the simple fact that the sphere is of aU configurations that
whose surface-area for a given volume is an absolute minimum, we
have seen it to be the one figure of equilibrium assumed by a drop
or vesicle when no disturbing factor is at hand; but such freedom
from counter-influences is likely to be rare, and neither does the rain-
drop nor the round world itself retain its primal sphericity. For one
thing, gravity will always be at hand to drag and distort our drop
or bubble, unless its dimensions be so minute that gravity becomes
insignificant compared with capillarity. Even the soap-bubble will
be flattened or elongated by gravity, according as we support it
from below or from above; and the bubble which is thinned out
almost to blackness will, from its small mass, be the one which
remains most nearly spherical*.

Innumerable new conditions will be introduced, in the shape of
comphcated tensions and pressures,, when one drop or bubble
becomes associated with another, and when a system of inter-
mediate films or partition-walls is developed between them. This
subject we shall discuss later, in connection with cell-aggregates or
tissues, and we shall find that further theoretical considerations are

* Cf. Dewar, On soap-bubbles of long duration, Proc. Roy. Inst. Jan. 19, 1929.

400 THE FORMS OF CELLS [ch.

needed as a preliminary to any such enquiry. Meanwhile let us
consider a few cases of the forms of cells, either sohtary, or in such
simple aggregates that their individual form is httle disturbed thereby.
Let us clearly understand that the cases we are about to consider
are those where the perfect sjnnmetry of the sphere is replaced by
another symmetry, less complete, such as that of an ellipsoidal or
cylindrical cell. The cases of asymmetrical deformation or dis-
placement, such as are illustrated in the production of a bud or
the development of a lateral branch, are much simpler; for here
we need only assume a slight and locahsed variation of surface-
tension, such as may be brought about in various ways through
the heterogeneous chemistry of the cell. But such diffused and
graded asymmetry as brings about for instance the ellipsoidal shape
of a yeast-cell is another matter.

If the sphere be the one surface of complete symmetry and
therefore of independent equilibrium, it follows that in every cell
which is otherwise conformed there must be some definite cause of
its departure from sphericity; and if this cause be the obvious one
of resistance offered by a solidified envelope, such as an egg-shell
or firm cell-wall, we must still seek for the deforming force which
was in action to bring about the given shape prior to the assumption
of rigidity. Such a cause may be either external to, or may lie
within, the cell itself. On the one hand it may be due to external
pressure or some form of mechanical restraint, as when we submit our
bubble to the partial restraint of discs or rings or more compHcated
cages of wire ; on the other hand it maybe due to intrinsic causes, which
must come under the head either of differences of internal pressure,
or of lack of homogeneity or isotropy in the surface or its envelope*.

* A case which we have not specially considered, but which may be found to
deserve consideration in biology, is that of a cell or drop suspended in a liquid of
varying density, for instance in the upper layers of a fluid (e.g. sea-water) at whose
surface condensation is going on, so as to produce a steady density-gradient. In
this case the normally spherical drop will be flattened into an oval form, with its
maximum surface-curvature lying at the level where the densities of the drop
and the surrounding liquid are just equal. The sectional outline of the drop has
been shewn to be not a true oval or ellipse, but a somewhat complicated quartic
curve. (Rice, Phil. Mag. Jan. 1915.) A more general case, which also may well
deserve consideration by the biologist, is that of a charged bubble in (for instance)
a uniform field of force : which will expand or elongate in the direction of the lines
of force, and become a spheroidal surface in continuous transformation with the
original sphere.

V] OF ASYMMETRY AND ANISOTROPY 401

Our formula of equilibrium, or equation to an elastic surface, is
P = jDg + (TjR + T' jR), where P is the internal pressure, jo^ any
extraneous pressure normal to the surface, R, R' the radii of
curvature at a point, and T, T' the corresponding tensions, normal
to one another, of the envelope.

Now in any given form which we seek to account for, R, R' are
known quantities; but all the other factors of the equation are
subject to enquiry. And somehow or other, by this formula, we
must account for the form of any solitary cell whatsoever (provided
always that it be not formed by successive stages of sohdification),
the cyhndrical cell of Spirogyra^ the elhpsoidal yeast-cell, or (as
we shall see in another chapter) even the egg of any bird. In
using this formula hitherto we have taken it in a simphfied form,
that is to say we have made several limiting assumptions. We have
assumed that P was the uniform hydrostatic pressure, equal in all
directions, of a body of hquid; we have assumed likewise that the
tension T was due to surface-tension in a homogeneous hquid film,
and was therefore equal in all directions, so that T = T' ; and we
have only dealt with surfaces, or parts of a surface, where extraneous
pressure, jo„, was non-existent. Now in the case of a bird's egg
the external pressure p^, that is to say the pressure exercised by
the walls of the oviduct, will be found to be a very important
factor; but in the case of the yeast-cell or the Spirogym, wholly
immersed in water, no such external pressure comes into play.
We are accordingly left in such cases as these last with 'two
hypotheses, namely that the departure from a spherical form is due
to inequahties in the internal pressure P, or else to inequahties in
the tension T, that is to say to a difference between T and T'.
In other words, it is theoretically possible that the oval form of a
yeast-cell is due to a greater internal pressure, a greater '"tendency
to grow" in the direction of the longer axis of the ellipse, or
alternatively, that with equal and symmetrical tendencies to growth
there is associated a difference of external resistance in respect of
the tension, and implicitly the molecular structure, of the cell- wall.
Now' the former hypothesis is not impossible. Protoplasm is far
from being a perfect fluid; it is the seat of various internal forces,
sometimes manifestly polar, and it is quite possible that the forces,
osmotic and other, which lead to an increase of the content of the

402 THE FORMS OF CELLS [ch.

cell and are manifested in pressure outwardly directed upon its wall
niay be unsymmetrical, and such as to deform what would otherwise
be a simple sphere. But while this hypothesis is not impossible,
it is not very easy of acceptance. The protoplasm, though not a
perfect fluid, has yet on the whole the properties of a fluid ; within
the small compass of the cell there is httle room for the development
of unsymmetrical pressures; and in such a case as Spirogyra, where
most part of the cavity is filled by watery sap, the conditions are
still more obviously, or more nearly, those under which a uniform
hydrostatic pressure should be displayed. But in variations of T,
that is to say of the specific surface-tension per unit area, we have
an ample field for all the various deformations with which we shall
have to deal. Our condition now is, that (TIR + T' jR) = a con-
stant; but it no longer follows, though it may still often be the
case, that this will represent a surface of absolute minimal area.
As soon as T and T' become unequal, we are no longer dealing
with a perfectly hquid surface film; but its departure from perfect
fluidity may be of all degrees, from that of a slight non-isotropic
viscosity to the state of a firm elastic membrane * ; and it matters
little whether this viscosity or semi-rigidity be manifested in the
self-same layer which is still a part of the protoplasm of the cell,
or in a layer which is completely differentiated into a distinct and
separate membrane. As soon as, by secretion or adsorption, the
molecular constitution of the surface-layer is altered, it is clearly
conceivable that the alteration, or the secondary chemical changes
which follow it, may be such as to produce an anisotropy, and to
render the molecular forces less capable in one direction than another
of exerting that contractile force by which they are striving to reduce
to a minimum the surface area of the cell. A slight inequality in
two opposite directions will produce the eUipsoid cell, and a great
inequahty will give rise to the cylindrical cell.

I take it therefore, that the cylindrical cell of Spirogyra, or any
other cyhndrical cell which grows in freedom from any manifest
external restraint, has assumed that particular form simply by
reason of the molecular constitution of its developing wall or

* Indeed any non-isotropic stiffness, even though T remained uniform, would
simulate, and be indistinguishable from, a condition of non-stiffness and non-
isotropic T.

v] OF ASYMMETRY AND ANISOTROPY 403

membrane; and that this molecular constitution was anisotropous,
in such a way as to render extension easier in one direction than
another. Such a lack of homogeneity or of isotropy in the
cell-wall is often rendered visible, especially in plant-cells, in the
form of concentric lamellae, annular and spiral striations, and the
like. But there exists yet another heterogeneity, to help us account
for the long threads, hairs, fibres, cylinders, which are so often
formed. Carl NageH said many years ago that organised bodies,
starch-grains, cellulose and protoplasm itself, consisted of invisible
particles, each an aggregate of many molecules — he called them
micellae; and these were isolated, or "dispersed" as we should say,
in a watery medium. This theory was, to begin with, an attempt to
account for the colloid state; but at the same time, the particles
were supposed to be so ordered and arranged as to render the
substance anisotropic, to confer on it vectorial properties as we say
nowadays, and so to account for the polarisation of light by a starch-
grain or a hair. It was so criticised by Biitschli and von Ebner that
it fell into disrepute, if not oblivion; but a great part of it was true.
And the micellar structure of wool, cotton, silk and similar substances
is now rendered clearly visible by the same X-ray methods as
revealed the molecular orientation, or lattice-structure, of a crystal
to von Laue.

It is now well known that the cell-wall has in many cases a definite structure
which depends on molecular assemblages in the material of which it is com-
posed, and is made visible by X-rays in the form of "diffraction patterns".
The green alga Valonia has very large bubbly cells, 2-3 centimetres long, with
cell-walls formed, as usual, of cellulose; this substance is a polysaccharide,
with long-chain molecules some 500 Angstrom-units, or say 0-05 /x long,
bound together sideways to form a multiple sheet or three-dimensional lattice.
In the cell-wall of Valonia one set of chains runs round in a left-handed
spiral, another forms meridians from pole to pole, and these two layers
are superposed alternately to build the wall. Hemp has two layers, both
running in right-handed spirals; flax two layers, crossing and recrossing in
spirals of opposite sign. Even the cytoplasm and its contents seem to be
influenced by molecular ''lignes directrices,'''' corresponding to the striae of
the cell-wall. Analogous but still more complicated results of molecular
structure are to be found in wool, cotton and other fibres*.

* Cf. R. D. Preston, Phil. Trans. (B), ccxxiv, p. 131, 1934: Preston and Astbury,
Proc. R.S. (B), cxxii, pp. 76-97, 1937; and many other important papers by
Astbury, van Iterson, Heyn, and others. We are brought by them to a borderland

404 THE FORMS OF CELLS [ch.

But this phenomenon, while it brings about a certain departure
from complete symmetry, is still compatible with, and coexistent
with, many of the phenomena which we have seen to be associated
with surface-tension. The symmetry of tensions still leaves the
cell a solid of revolution, and its surface is still a surface of equi-
librium. The fluid pressure within the cy Under still causes the
film or membrane which caps its ends to be of a spherical form.
And in the young cell, where the surface pellicle is absent or but
little differentiated, as for instance in the oogonium of Aclilya or
in the young zygospore of Spirogyra, we see the tendency of the
entire structure towards a spherical form reasserting itself: unless,
as in the latter case, it be overcome by direct compression within
the cyhndrical mother-cell. Moreover, in those cases where the
adult filament consists of cylindrical cells we see that the young
germinating spore, at first spherical, very soon assumes with growth
an elliptical or ovoid form — the direct result of an incipient aniso-
tropy of its envelope, which when more developed will convert the
ovoid into a cyhnder. We may also notice that a truly cyhndrical
cell is comparatively rare, for in many cases what we call a
cylindrical cell shews a distinct bulging of its sides; it is not truly
a cyhnder, but a portion of a spheroid or ellipsoid.

Unicellular organisms in general — protozoa, unicellular crypto-
gams, various bacteria and the free isolated cells, spores, ova, etc.
of higher organisms — are referable for the most part to a small
number of typical forms ; but there are many others in which either
no symmetry is to be recognised, or in which the form is clearly
not one of equihbrium. Among these latter we have Amoeba itself
and all manner of amoeboid organisms, and also many curiously
shaped cells such as the Trypanosomes and various aberrant
Infusoria. We shall return to the consideration of these; but in
the meanwhile it will suffice to say (and to repeat) that, inasmuch
as their surfaces are not equihbrium-surfaces, so neither are the
Uving cells themselves in any stable equihbrium. On the contrary,
they are in continual flux and movement, each portion of the

between chemical and histological structure, where micellae and long- chain molecules
enlarge and alter our conceptions not only of cellulose and keratin, but of pseudopodia
and cilia, of bone and muscle, and of the naked surface of the cell. See L. E. R.
Picken, The fine structure of biological systems, Biol. Reviews, xv, pp. 133-67, 1940.

V] OF STABLE AND UNSTABLE EQUILIBRIUM 405

surface constantly changing its form, passing from one phase to
another of an equihbrium which is never stable for more than a
moment, and which death restores to the stable equilibrium of a
sphere. The former class, which rest in stable equihbrium, must
fall (as we have seen) into two classes — those whose equilibrium
arises from liquid surface-tension alone, and those in whose con-
formation some other pressure or restraint has been superimposed
upon ordinary surface-tension.

To the fact that all these organisms belong to an order of
magnitude in which form is mainly, if not wholly, conditioned and
controlled by molecular forces is due the hmited range of forms
which they actually exhibit. They vary according to varying
physical conditions. Sometimes they do so in so regular and
orderly a way that w^e intuitively explain them as "phases of a
Hfe-history," and leave physical properties and physical causation
alone: but many of their variations of form we treat as exceptional,
abnormal, decadent or morbid, and are apt to pass these over in
neglect, while we give our attention to what we call a typical or
"characteristic" form or attitude. In the case of the smallest
organisms, bacteria, micrococci, and so forth, the range of form is
especially limited, owing to their minuteness, the powerful pressure
which their highly curved surfaces exert, and the comparatively
homogeneous nature of their substance. But within their narrow
range of possible diversity these minute organisms are protean in
their changes of form. A certain species will not only change its
shape from stage to stage of its little "cycle" of hfe; but it will
be remarkably different in outward form according to the circum-
stances under which we find it, or the histological treatment to
which we subject it. Hence the pathological student, commencing
the study of bacteriology, is early warned to pay little heed to
differences of form, for purposes of recognition or specific identi-
fication. Whatever grounds we may have for attributing to
these organisms a permanent or stable specific identity (after
the fashion of the higher plants and animals), we can seldom
safely do so on the ground of definite and always recognisable
form: we may often be inclined, in short, to ascribe to them a
physiological (sometimes a "pathogenic") rather than a morpho-
logical specificity.

406

THE FORMS OF CELLS

[CH.

Many unicellular forms, and a few other simple organisms, are
spherical, and serve to illustrate in the simplest way the point at
issue. Unicellular algae, such as Protococcus or Halisphaera, the
innumerable floating eggs of fishes, the floating unilocular foraminifer
Orbulina, the lavely green multicellular Volvox of our ponds, all
these in their several grades of simplicity or complication are so
many round drops, spherical because no alien forces have deformed
or mis-shapen them. But observe that, with the exception of
Volvox, whose spherical body is covered wholly and uniformly with
minute ciha, all the above are passive or inactive forms; and in a
"resting" or encysted phase the spherical form is common and
general in a great range of unicellular organisms.

Conversely, we see that those unicellular forms which depart
jUBrkedly from sphericity — excluding for the moment the amoeboid

forms and those provided with skeletons
— are all cihate or flagellate. Ciha and
flagella are sui generis ; we know nothing
of them from the physical side, we cannot
reproduce or imitate them in any non-
hving drop or fluid surface. But we can
easily see that they have an influence on
form, besides serving for locomotion.
When our httle Monad or Euglena
develops a flagellum, that is in itself an
indication of asymmetry or "polarity,"
of non-homogeneity of the little cell ; and
in the various flagellate types the flagellum or its analogues always
stand on prominent points, or ends, or edges of the cell — on parts,
that is to say, where curvature is high and surface-tension may be
expected to be low — for the product of surface-tension by mean
curvature tends to be constant.

Fig. 123. A flagellate "monad,"
Distigma proteus Ehr,
After Saville Kent.

The minute dimensions of a cilium or a flagellum are such that the molecular
forces leading to surface-tension must here be under peculiar conditions and
restraints; we cannot hope to understand them by comparison with a whip-
lash, or through any other analogy drawn from a different order of magnitude.
I suspect that a ciliary surface is always electrically charged, and that
a point- charge is formed or induced in each cilium or flagellum. Just as we
learn the properties of a drop or a jet as phenomena proper to their scale of
magnitude, so some day we shall learn the very different physical, but

v] OF CILIA AND FLAGELLA 407

microcosmic, properties of these minute, mobile, pointed, fluid or semi-fluid
threads.*

Cilia, like flagella, tend to occupy positions, or cover surfaces,
which would otherwise be unstable; and often indeed (as in a
trochosphere larva or even in a Rotifer) a ring of cilia seems to
play the very part of one of Plateau's wire rings, supporting and
steadying the semi-fluid mass in its otherwise unstable configura-
tion. Let us note here (in passing) what seems to be an analogous
phenomenon. Chitinous hairs, spines or bristles are common and
characteristic structures among the smaller Crustacea, and more or
less generally among the Arthropods. We find them at every
exposed point or corner; they fringe the sharp edge or border of
a limb; as we draw the creature, we seem to know where to put
them in ! In short, they tend to occur, as the flagella do, just where
the surface-tension would be lowest, if or when the surface was in
a fluid condition.

Of the other surfaces of Plateau, we find cylinders enough and
to spare in Spirogyra and a host of other filamentous algae and
fungi. But it is to the vegetable kingdom that we go to find them,
where a cellulose envelope enables the cyUnder to develop beyond
its ordinary limitations.

The unduloid makes its appearance whenever sphere or cyhnder
begin to give way. We see the transitory figure of an unduloid in
the normal fission of a simple cell, or of the nucleus itself; and we
have already seen it to perfection in the incipient headings of a
spider's web, or of a pseudopodial thread of protoplasm. A large
number of infusoria have unduloid contours, in part at least; and
this figure appears and reappears in a great variety of forms. The
cups of various Vorticellae (Fig. 124), below the ciliated ring, look
like a beautiful series of unduloids, in every gradation of form, from
what is all but cylindrical to all but a perfect sphere; moreover
successive phases in their hfe-history appear as mere graded changes

* It is highly characteristic of a cilium or a flagellum that neither is ever seen
motionless, unless the cell to which it belongs is moribund. "I believe the motion
to be ceaseless, unconscious and uncontrolled, a direct function of the chemical
and physical environment "; tieorge Bidder, in Presidential Address to Section D,
British Associqtion, 1927. Cf. also James Gray, Proc. R.S. (B), xcix, p. 398,
1926.

408

THE FORMS OF CELLS

[CH.

of unduloid form. It has been shewn lately, in one or two
instances at least, that species of Vorticella may "metamorphose"
into one another : in other words, that contours supposed to charac-
terise species are not "specific" These VorticelUd unduloids are

>^-

Fig. 124. Various species of Vorticella.

M

Fig. 1 25. Various species of Salpingoeca.

Fig. 126. Various species of Tintinnus, Dinohryon and Codonella.
After Saville Kent and others.

not fully symmetrical ; rather are they such unduloids as develop
when we suspend an oil-globule between two unequal rings, or blow
a bubble between two unequal pipes. For our Vorticellid bell hangs
by two terminal supports, the narrow stallj to which it is attached
below, and the thickened ring from which spring its circumoral
cilia; and it is most interesting to see how, when the bell leaves

OF VARIOUS UNDULOIDS

409

its stalk (as sometimes happens) and swims away, a new ring of
cilia comes into being, to encircle and support its narrow end.

Similar unduloids may be traced in even greater variety among
other famihes or genera of the Infusoria. Sometimes, as in Vorticella
itself, the unduloid is seen in the contour of the soft semifluid
body of the hving animal. At other times, as in Salpingoeca,
Tintinnus, and many other genera, we have a membranous cup
containing the animal, but originally secreted by, and moulded
upon, its semifluid hving surface. Here we have an excellent
illustration of the contrast between the different ways in which
such a structure may be regarded and interpreted. The teleological
explanation is that it is developed for the sake of protection,

127. Vaginicola.

Fig. 128. FolUculina.

as a domicile and shelter for the httle organism within. The
mechanical explanation of the physicist (seeking after the "efficient,"
not the "final" cause) is that it owes its presence, and its actual
conformation, to certain chemico-physical conditions: that it was
inevitable, under the given conditions, that certain constituent
substances present in the protoplasm should be drawn by molecular
forces to its surface layer; that under this adsorptive process, the
conditions continuing favourable, the particles accumulated and
concentrated till they formed (with the help of the surrounding
medium) a pellicle or membrane, thicker or thinner as the case
might be; that this surface pellicPe or membrane was inevitably
bound, by molecular forces, to contract into a surface of the
least possible area which the circumstances permitted; that in
the present case the symmetry and "freedom" of the system
permitted, and ipso facto caused, this surface to be a surface of
revolution; and that of the few surfaces of revolution which, as

410

THE FORMS OF CELLS

[CH.

being also surfaces minimae areae, were available, the unduloid was
manifestly the one permitted, and ipso facto caused, by the dimen-
sions of the organism and other circumstances of the case. And
just as the thickness or thinness of the pellicle
was obviously a subordinate matter, a mere
matter of degree, so we see that the actual
outhne of this or that particular unduloid is
also a very subordinate matter, such as physico-
chemical variants of a minor order would suffice
to -bring about; for between the various undu-
loids which the various species of Vorticella
represent, there is no more real difference than
that difference of ratio or degree which exists
between two circles of different diameter, or
two hnes of unequal length.

In many cases (of which' Fig. 129 is an
example) we have a more or less unduloid form
exhibited not by a surrounding pelHcle or shell,
but by the soft protoplasmic body of a ciliated
organism; in such cases the form is mobile,
and changes continually from one to another
unduloid contour according to the movements
of the animal.* We are deahng here with no
stable equihbrium, but possibly with a subtle
problem of ''stream-lines," as in the difficult
but beautiful problems suggested by the form
of a fish. But this whole class of cases, and
of problems, we merely take note of here; we shall speak of them
again, but their treatment is hard.

In considering such series of forms as these various unduloids we
are brought sharply up (as in the case of our bacteria or micrococci)
against the biological concept of organic species. In the intense
classificatory activity of the last hundred years it has come about
that every form which is apparently characteristic, that is to Say
which is capable of being described or portrayed, and of being

* Doflein lays stress, in like manner, on the fact that Spirochade, unlike
Spirillum, "ist nicht von einer starren Membran umhiillt," and that waves of
contraction may be seen passing down its body.

Fig. 129. Trachelo-
phyllum. After
Wreszniowski.

V] OF SNOW-CRYSTALS 411

recognised when met with again, has been recorded as a species —
for we need not concern ourselves with the occasional discussions,
or individual opinions, as to whether such and such a form deserves
"specific rank," or be "only a variety." And this secular labour
is pursued in direct obedience to the precept of the Systema Naturae
— ''ut sic in summa confusione rerum apparenti, summus conspiciatur
Naturae ordo.'' In like manner the physicist records, and is entitled
to record, his many hundred "species" of snow-crystals*, or of
crystals of calcium carbonate. Indeed the snow-crystal illustrates to
perfection how Nature rings the changes on every possible variation
and permutation and combination of form: subject only to the
condition (in this instance) that a snow-crystal shall be a plane,
symmetrical, rectilinear figure, with all its external angles those of
a regular hexagon. We may draw what we please on a sheet of
"hexagonal paper," keeping to its lines; and when we repeat our
drawing, kaleidoscope-fashion, about a centre, the stellate figure so
obtained is sure to resemble one or another of the many recorded
species of snow-crystals. And this endless beauty of crystalline
form is further enhanced when the flakes begin to thaw, and all
their feathery outlines soften. But regarding these "species" of his,
the physicist makes no assumptions: he records them simpliciter;
he notes, as best he can, the circumstances (such as temperature or
humidity) under which each occurs, in the hope of elucidating the.
conditions which determine their formation f; but above all, he

* The case of the snow-crystals is a particularly interesting one; for their
"distribution" is analogous to what we find, for instance, among our microscopic
skeletons of Radiolarians. That is to say, we may one day meet with myriads
of some one particular form or species, and another day with myriads of another
while at another time and place we may find species intermingled in all but
inexhaustible variety. Cf. e.g. J. Glaisher, Illustrated London News, Feb. 17, 1855
Q.J. M.S. m, pp. 179-185, 1855; Sir Edward Belcher, Last of the Arctic Voyages,
II, pp. 288-306 (4 plates), 1855; William Scoresby, An Account of the Arctic Regions,
Edinburgh, 1820; G. Hellmann, Schneekrystalle, Berlin, 1893; Bentley and Hum
phreys. Snow Crystals, New York, 1931; and the especially beautiful figures of
Nakaya and Hasikura in Journ. Fac. Sci. Hokkaido, Dec. 1934.

t Every snow-crystal tells, more or less plainly, the story of its own development.
The cold upper air is saturated with water- vapour, but this is scanty and rarefied
compared with the space in which snow- crystallisation is going on. Hence
crystallisation tends to proceed only along the main axes, or cardinal framework,
of the crystalline structure of ice ; in so doing it gives a visible picture or actual
embodiment of the trigonal-hexagonal space-lattice, in the endless permutations
and combinations of its constituent elements.

412 THE FORMS OF CELLS [ch.

does not introduce the element of time, and of succession, or discuss
their origin and affihation as an historical sequence of events. But
in biology, the term species carries with it many large though often
vague assumptions; though the doctrine or concept of the "per-
manence of species" is dead and gone, yet a certain quasi-permanency
is still connoted by the term. If a tiny foraminiferal shell, a Lagena
for instance, be found hving to-day, and a shell indistinguishable
from it to the eye be found fossil in the Chalk or some still more
remote geological formation, the assumption is deemed legitimate
that that species has "survived," and has handed down its minute
specific character or characters from generation to generation,
unchanged for untold milHons of years*. If the ancient forms be
Hke to rather than identical with the recent, we still assume an
unbroken descent, accompanied by the hereditary transmission of
common characters and progressive variations. And if two identical
forms be discovered at the ends of the earth, still (with occasional
slight reservations on the score of possible "homoplasy") we build
hypotheses on this fact of identity, taking it for granted that the
two appertain to a common stock, whose dispersal in space must
somehow be accounted for, its route traced, its epoch determined,
and its causes discussed or discovered. In short, the naturaUst
admits no exception to the rule that a natural classification can only
be a genealogical one, nor ever doubts that '"The fact that we are able
to classify organisms at all in accordance with the structural charac-
teristics which they present is due to the fact of their being related by
descenf\'' But this great and valuable and even fundamental
generahsation sometimes carries us too far. It may be safe and
sure and helpful and illuminating when we apply it to such complex
entities — such thousand-fold resultants of the combination and
permutation of many variable characters — as a horse, a lion or an
eagle; but (to my mind) it has, a very different look, and a far less
firm foundation, when we attempt to extend it to minute organisms
whose specific characters are few and simple, whose simplicity

* Cf. Bergson, Creative Evolution, p. 107: "Certain Foraminifera have not
varied since the Silurian epoch. Unmoved witnesses of the innumerable revolu-
tions that have upheaved our planet, the Lingulae are today what they were at
the remotest times of the palaeozoic era."

t Ray Lankester, A.M.N.H. (4), xi, p. 321, 1873.

V] OF FORM AND SPECIES 413

becomes more manifest from the point of view of physical and
mathematical analysis, and whose form is referable, or largely
referable, to the direct action of a physical force. When we come
to the minute skeletons of the Radiolaria we shall again find our-
selves dealing with endless modifications of form, in which it becomes
more and more difficult to discern, and at last vain and hopeless to
apply, the guiding principle of affihation or '^phylogeny.''

Among the Foraminifera we have an immense variety of forms,
which, in the hght of surface-tension and of the principle of minimal
area, are capable of explanation and of reduction to a small number
of characteristic types. Many of them are composite structures,
formed by the successive imposition of cell upon cell, and these we shall
deal with later on ; let us glance here at the simpler conformations
exhibited by the single chambered or " monothalamic " genera, and
perhaps one or two of the simplest composites.

We begin with forms like Astrorhiza (Fig. 320, p. 703), which are
large, coarse and highly irregular, and end with others which are
minute and delicate, and which manifest a perfect and mathe-
matical regularity. The broad difference between these two types
is that the former are characterised, like Amoeba, by a variable
surface-tension, and consequently by unstable equihbrium; but the
strong contrast between these and the regular forms is bridged over
by various transition-stages, or differences of degree. Indeed, as
in all other Rhizopods, the very fact of the emission of pseudopodia,
which are especially characteristic of this group of animals, is
a sign of unstable surface-equihbrium ; and we must therefore
consider, or may at least suspect, that those forms whose shells
indicate the most perfect symmetry and equilibrium have secreted
these during periods when rest and uniformity of surface-conditions
contrasted with the phases of pseudopodial activity. The irregular
forms are in almost all cases arenaceous, that is to say they have
no soHd shells formed by steady adsorptive secretion, but only a
looser covering of sand grains with which the protoplasmic body
has come in contact and cohered. Sometimes, as in Ramulina, we
have a calcareous shell combined with irregularity of form; but
here we can easily see a partial and as it were a broken regularity,
the regular forms of sphere and cylinder being repeated in various

414

THE FORMS OF CELLS

[CH.

parts of the ramified mass. When we look more closely at the
arenaceous forms, we find the same thing true of them; they
represent, in whole or part, approximations to the surfaces of
equilibrium, spheres, cylinders and so forth. In Aschemonella we
have a precise rephca of the calcareous Ramuliyia; and in Astrorhiza
itself, in the forms distinguished by jiaturalists as A. crassatina,
what is described as the " subsegmented interior*" seems to shew

Fig. 130. Various species of Zagrew«. After Brady.

the natural, physical tendency of the long semifluid cylinder of
protoplasm to contract at its limit of stability into unduloid
constrictions, as a step towards the breaking up into separate
spheres : the completion of which process is restrained or prevented
by contact with the unyielding arenaceous covering.

Passing to the typical calcareous Foraminifera, we have the most
symmetrical of all possible types in the perfect sphere of Orbulina ;
this is a pelagic organism, whose floating habitat gives it a field of

♦ Brady, Challenger Monograph, pi. xx, p. 233.

V] OF HANGING DROPS 415

force of perfect symmetry. Save for one or two other forms which
are also spherical, or approximately so, like Thurammina, the rest
of the monothalamic calcareous
Foraminifera are all comprised by
naturalists within the genus
Lagena. This large and varied
genus consists of "flask-shaped"
shells, whose surface is that of an
unduloid, or, hke that of a flask
itself, an unduloid combined with
a portion of a sphere. We do
not know the circumstances under
which the shell oi Lagena is formed,
nor the nature of the force by
which, during its. formation, the
surface is stretched out into the
unduloid form; but we may be
pretty sure that it is suspended
vertically in the sea, that is to
say in a position of symmetry as
regards its vertical axis, about
which the unduloid surface of re-
volution is symmetrically formed.

4 5 6

Fig. 131. Roman pottery, for comparison

with species oi Lagena. E.g., 1, 2, with

L. sulcata; 3, L. orbignyana; 4, L.

striata; 5, L. crenata; 6, L. stelligera.

At the same time we have other
types of the same shell in which the form is more or less flattened ;
and these are doubtless the cases in which such symmetry of position
was not present, or was replaced by a broader, lateral contact with
the surface pellicle*.

While Orhulina is a simple spherical drop, Lagena suggests to our
minds a hanging drop, drawn out to a longer or shorter neck by

* That the Foraminifera not only can but do hang from the surface of the
water is confirmed by the following apt quotation which I owe to Mr E. Heron -
Allen: "Quand on place, comme il a ete dit, le depot provenant du lavage des
fucus dans un fiacon que Ton remplit de nouvelle eau, on voit au bout d'une heure
environ les animaux [Gromia dujardinii] se mettre en mouvement et commencer
a grimper. Six heures apres ils tapissent Texterieur du flacon, de sorte que les plus
eleves sont a trente-six ou quarante-deux millimetres du fond; le lendemain
beaucoup d'entre eux, apres avoir atteint le niveau du liquide, ont continue a rarnper
a sa surface, en se laissant pendre au-dessous comme certains mollusques gastero-
podes." (F. Dujardin, Observations nouvelles sur les pretendus cephalopodes
microscopiques, Ann. des Sci. Nat. (2), iii, p. 312, 1835.)

416 THE FORMS OF CELLS [ch.

its own weight, aided by the viscosity of the material. Indeed the
various hanging drops, such as Mr C. R. DarUng shews us, are the
most beautiful and perfect unduloids, with spherical ends, that it
is possible to conceive. A suitable hquid, a httle denser than water
and incapable of mixing with it (such as ethyl benzoati), is poured
on a surface of water. It spreads over the surface and gradually
forms a hanging drop, approximately hemispherical; but as more
liquid is added the drop sinks or rather stretches downwards, still
adhering to the surface fihn; and the balance of forces between
gravity and surface-tension results in the unduloid contour, as the
increasing weight of the drop tends to stretch it out and finally
Sreak it in two. At the moment of rupture, by the way, a tiny

Fig. 132. Large "hanging drops" ot oil. After Darling.

droplet js formed in the attenuated neck, such as we described in
the normal division of a cyhndrical thread.

The thin, fusiform, pointed, non-globular Lagenas are less easily
explained. Surface-tension, which tends to keep the drop spherical,
is overmastered here, and the elongate shape suggests the viscous
drag of a shearing fluid*.

To pass to a more highly organised class of animals, v^e find the unduloid
beautifully exemplified in the little flask-shaped shells of certain Pteropod
mollusca, e.g. Cuvierina-\. Here again the symmetry of the figure would
at once lead us to suspect that the creature lived in a position of symmetry
to the surrounding forces, as for instance if it floated in the ocean in an
erect position, that is to say with its long axis coincident with the direction
of gravity; and this we know to be actually the mode of life of the little
Pteropod.

* Cf. G. I. Taylor, The formation of emulsions in definable fields of flow, Proc. R.S.
(A), No. 858, p. 501, 1934.
t Cf. Boas, Spolia Atlantica, 1886, pi. 6.

V] OF RETICULATE OR WRINKLED CELLS 417

Many species of Lagena are complicated and beautified by a
pattern, and some by the superaddition to the shell of plane
extensions or "wings." These latter give a secondary, bilateral
symmetry to the little shell, and are strongly suggestive of a phase
or period of growth in which it lay horizontally on the surface,
instead of hanging vertically from the surface-film : in which, that
is to say, it was a floating and not a hanging drop. The pattern
is of two kinds. Sometimes it consists of a sort of fine reticulation,
with rounded or more or less hexagonal interspaces: in other cases
it is produced by a symmetrical series of ridges or folds, usually
longitudinal, on the body of the flask-shaped cell, but occasionally
transversely arranged upon the narrow neck. The reticulated and
folded patterns we may consider separately. The netted pattern
is very similar to the wrinkled surface of a dried pea, or to the more
regular wrinkled patterns on poppy and other seeds and even pollen-
grains. If a spherical body after developing a "skin" begin to
shrink a little, and if the skin have so far lost its elasticity as to
be unable to keep pace with the shrinkage of the inner mass, it will
tend to fold or wrinkle; and if the shrinkage be uniform, and the
elasticity and flexibihty of the skin be also uniform, then the amount
of foldings will be uniformly distributed over the surface. Little
elevations and depressions will appear, regularly interspaced, and
separated by concave or convex folds. These being of equal size
(unless the system be otherwise perturbed), each one wiU tend to
be surrounded by six others; and when the process has reached its
limit, the intermediate boundary-walls, or folds, will be found
converted into a more or less regular pattern of hexagons. To these
symmetrical wrinkles or shrinkage-patterns we shall return again.

But the analogy of the mechanical wrinkHng of the coat of a
seed is but a rough and distant one; for we are deahng with
molecular rather than with mechanical forces. In one of Darling's
experiments, a little heavy tar-oil is dropped on to a saucer of
water, over which it spreads in a thin film shewing beautiful
interference colours after the fashion of those of a soap-bubble.
Presently tiny holes appear in the film, which gradually increase
in size till they form a cellular pattern or honeycomb, the oil
gathering together in the meshes or walls of the cellular net. Some
action of this sort is in aU probability at work in a surface-film

418 THE FORMS OF CELLS [ch.

of protoplasm covering the shell. As a physical phenomenon the
actions involved are by no means fully understood, but surface-
tension, diffusion and cohesion play their respective parts therein*.
The very perfect cellular patterns obtained by Leduc (to which we
shall have occasion to refer in a subsequent chapter) are diffusion
patterns on a larger scale, but not essentially different.

The folded or pleated pattern is doubtless to be explained, in

a general way, by the shrinkage of a surface-film under certain

conditions of viscous or frictional restraint.

A case which (as it seems to me) is closely

alhed to that of our foraminiferal shells is

described by Quincke t, who let a film of

chromatised gelatin or of resin set and harden

upon a surface of quicksilver, and found that

the little solid pelHcle had been thrown into

a pattern of symmetrical folds, as fine as a

Fig.^ 133. diffraction grating. If the surface thus folded

or wrinkled be a cyhnder, or any other figure with one principal axis

* This cellular pattern would seem to be related to the "cohesion figures"
described by Tomlinson in various surface-films {Phil. Mag. 1861-70); to the
"tesselated struct